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On unique solvability of nonlocal drift–diffusion-type problems
Nonlinear Analysis 56 (2004) 803 – 830
www.elsevier.com/locate/na
On unique solvability of nonlocal drift–di#usion-type problems H. Gajewskia;∗ , I.V. Skrypnikb a Weierstra-Institute
b Institute
for Applied Analysis and Stochastics, Mohrenstr. 39, Berlin D-10117, Germany for Applied Mathematics and Mechanics, Rosa Luxemburg Str. 74, 340114, Donetsk, Ukraine Received 14 May 2003; accepted 21 October 2003
Abstract We prove a priori estimates in L2 (0; T ; W 1; 2 ()) and L∞ (QT ), existence and uniqueness of solutions to Cauchy–Neumann problems for parabolic equations n @ @(u) @(u − v) (u) bi t; x; + a(t; x; v; u) = 0; (0.1) − @x @xi @t i=1 (t; x) ∈ QT = (0; T ) × ⊂ Rn+1 , where (u) = @(u)=@u ¿ 0 and the function v is de8ned by the nonlocal expression v(t; x) = − K(x; y)[(u(t; y)) − f(t; y)] dy; (0.2)
instead of solving an elliptic boundary problem as in the corresponding local case. Such problems arise as mathematical models of various di#usion–drift processes driven by gradients of local particle concentrations and nonlocal interaction potentials. An example is the transport of electrons in semiconductors, where u has to be interpreted as chemical and v as electro-statical potential. ? 2003 Elsevier Ltd. All rights reserved. MSC: 35B45; 35K15; 35K20; 35K65 Keywords: Nonlinear parabolic equations; Nonlocal drift; Bounded solutions; Uniqueness; Nonstandard assumptions; Degenerate type
∗
Corresponding author. E-mail address: [email protected] (H. Gajewski).
0362-546X/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2003.10.014
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1. Introduction We prove a priori estimates, existence and uniqueness of weak solutions to initial boundary value problems of the form n @(u − v) @ @(u) − + a(t; x; v; u) = 0 (t; x) ∈ QT ; (u)bi t; x; (1.1) @t @xi @x i=1 v(t; x) = − K(x; y)[(u(t; y)) − f(t; y)] dy (t; x) ∈ QT ; (1.2) n i=1
bi t; x;
@(u − v) @x
cos (; xi ) = 0
(t; x) ∈ = (0; T ) × @;
(1.3)
u(0; x) = h(x); x ∈ ; (1.4) u where (u) = 0 (s) ds, (u) ¿ 0, is a bounded open set in Rn and QT = (0; T ) × , T ¿ 0. In the case of smooth boundary @ of the set , is the outer unit normal on @ and (; xi ) is the angle between and the xi -axis. In the physical motivation and derivation (cf. [1,7]) of systems like (1.1)–(1.3) the ‘free energy’ c(t; y) −1 − f(t; y) dy d x; ( (c)) + c K(x; y) F(c) = 2 u (u) = s(s) ds (1.5) 0
plays an important role. Here c=(u) and ◦−1 can be seen as particle concentration and Legendre transformed of the primitive of , respectively, whereas the respective integrands model local and nonlocal particle interaction. Then, provided the reaction term a vanishes, system (1.1)–(1.4) describes the mass conserving evolution of c from the initial value c0 =(h) towards critical points or even minimizers of F under di#usion and drift forces, caused by the local and the global term in F, respectively. Moreover, the functional F will be also the key for our mathematical analysis of the system (cf. Theorem 1). Indeed, in the case that: the kernel K is symmetrical, the vector 8eld {bi (t; x; ·)} ∈ (Rn → Rn ) is monotone and bi (t; x; 0) = a = 0, we 8nd for solutions u; v of (1.1)–(1.3) according to the second law of thermodynamics @(u) dF(c) = (u − v) d x dt @t n @(u − v) @(u − v) bi t; x; d x 6 0; = − (u) @x @xi i=1
that means, F is Lyapunov functional in that case. Thus problems of the form (1.1)–(1.4) arise as intrinsically nonlocal mathematical models of various applied problems, for instance reaction–drift–di#usion processes of
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electrically charged species, phase transition processes and transport processes in porous media. The investigation of nonlinear nonlocal problems has received much attention in last years. In particular, in the papers [7,8,12,13] nonlocal models of phase separation were formulated and studied. One way to pass from (1.1)–(1.4) to local models consists in replacing the integral equation (1.2) by an elliptic di#erential equation like n @v @ (x) + (u) = f(t; x) (t; x) ∈ QT ; (1.6) − @xi @xi i=1
completed by some boundary condition for the function v. Such local problems were studied by many authors (cf. [6,10]). See also the papers [2,3,16], where degenerate parabolic equations were studied. Most strong results for local drift–di#usion type problems have been recently proved in [7]. Another way, applied in phase separation processes (cf. [4]), consists in replacing (1.5) by a local Ginsburg–Landau free energy expression 2
@c −1 F(c) = ( (c)) + c(1 − c) + d x; ; positive constants 2 @x and results in a fourth-order Cahn–Hilliard equation instead of the nonlocal second-order system (1.1)–(1.2). We consider problem (1.1)–(1.4) under standard conditions for the functions bi and some conditions for the function a to be formulated in Section 2. Our main speci8c assumption concerning Eq. (1.1) reads: (1 ) ∈ (R1 → R1 ) with (u) ¿ 0, u ∈ R1 , is continuous and has a piecewise continuous derivative such that (u)=(u) is nonincreasing on R1 . This condition seems natural in view of properties of probability particle distribution functions arising in mathematical physics. So in the semiconductor theory [1,6] relevant examples for functions satisfying condition (1 ) are given by = F!+1 , = = F! , where F! denotes the Fermi integral ∞ s! ds 1 F! (u) = ; ! ¿ − 1: (1.7) (! + 1) 0 1 + exp (s − u) Another example comes from phase separation problems [5,11], where the Fermi function 1 1 ; (1.8) ; (u) = (u) = (u) = −u u 1+e (1 + e )(1 + e−u ) plays a role corresponding to that of F!+1 . Our main assumption on the kernel K(x; y) is (K1 ) The function K(x; y) is de8ned for x; y ∈ and symmetrical, i.e., K(x; y) = K(y; x). Moreover, K(·; y) ∈ W 1; 1 () for almost every y ∈ such that @K(x; y) dy + ess sup @K(x; y) d x 6 –: (1.9) |K(x; y)| + ess sup @x @x x∈ y∈
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We remark that condition (K1 ) implies (cf. Lemma 1 below) properties as assumed in [5,11] for integral operators generated by kernels K(x; y) = K(|x − y|). Let us also remark that kernels |x − y|2−n ; log 1=|x − y|, corresponding to Newton potentials and fundamental solutions of Eq. (1.6) with bounded measurable function , satisfy condition (K1 ) [15]. The Green function for Eq. (1.6) satis8es condition (K1 ) in the cases of Dirichlet or Neumann boundary conditions for suMcient smooth @ and . Conditions on guarantying condition (K1 ) for the Green function can be formulated also in terms of smallness of the number −1 ess sup (x) ess inf (x) − 1: x∈
x∈
We formulate our assumptions and main results in Section 2. First, a priori estimates for the solution (u; v) are given in Section 3. In that section we prove also regularity properties of the function v, important for further considerations. An estimate of u in L∞ (QT ) is given in Section 4. Sections 5 and 6 are devoted to proofs of existence and uniqueness of solutions to problem (1.1)–(1.4) respectively. We are planning in forthcoming papers to apply our approach to systems of equations describing reaction–drift–di#usion processes in isothermal and non-isothermal cases. 2. Formulation of assumptions and main results Let be a bounded open set in Rn and QT = (0; T ) × , T ¿ 0. We assume that n ¿ 2. For n 6 2 it is necessary to make simple changes in our conditions that are connected with Sobolev’s embedding theorem. We assume the following condition on the set : (@) is such that the embeddings W 1; 1 () ⊂ Ln=(n−1) (), W 1; p () ⊂ L∞ () hold for p ¿ n. In view of the proof of a priori estimates for solutions of problem (1.1)–(1.4) we need restrictions on growth and on degeneration of the function as u → ±∞. From condition (1 ) the existence of $± = lim (u)
(2.1)
u→±∞
follows. For nonconstant functions (u) at least one of the limits $− , $+ is zero [8]. Studying the problem (1.1)–(1.4) we have to distinguish the cases of zero or non-zero values of $± . Therefore we shall consider two cases: ($1 ) $− = 0; $+ = 0;
($2 ) $− = $+ = 0:
Note that examples for ($1 ) and ($2 ) are given by (1.7) and (1.8), respectively. Our additional restrictions on the function are following: (2 ) if condition ($1 ) holds then positive constants 1 ; ! exist such that 2 ! ! −1 u ¿ 0; 0 6 ! ¡ ; 1 (u + 1) 6 (u) 6 1 (u + 1); n−1
(2.2)
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(3 ) there exists a positive constant 2 such that | (u)| 6 2 (u)
(2.3)
for u ¡ 0 in the case of condition ($1 ) and ∀u ∈ R1 if condition ($2 ) holds. Remark 1. The Fermi integrals F! (1.7) satisfy condition (2 ) for −1¡!¡(3 − n)= (n−1) and in particular for the physically most relevant case !=−1=2. In semiconductor modelling, especially in the classical drift–di#usion model, Fermi statistics is often approximated by Boltzmann statistics, i.e., F! is replaced by the exponential function. But, that approach violates condition 2 and leads to problems in proving uniqueness results at least in the three-dimensional case n = 3 and nonsmooth data. Let the functions a; bi from (1.1) satisfy the assumptions: (i) a(t; x; v; u), bi (t; x; &), i = 1; : : : ; n, are measurable with respect to t; x for every u; v ∈ R1 , & ∈ Rn and continuous with respect to u; v ∈ R1 , & ∈ Rn , for almost every (t; x) ∈ QT ; bi (t; x; 0) = 0, i = 1; : : : ; n; (ii) there exist positive constants 1 ; 2 such that ∀& ; & ∈ Rn and (t; x) ∈ QT n
[bi (t; x; & ) − bi (t; x; & )](&i − &i ) ¿ 1 |& − & |2 ;
i=1
|bi (t; x; &)| 6 2 (|&| + 1);
i = 1; : : : ; n;
(iii) there exist nonnegative functions $1 ∈ L1 (QT ), $ ∈ Lp1 (QT ), p1 ¿ (n + 2)=2, such that for arbitrary (t; x) ∈ QT , v; u ∈ R1 a(t; x; v; u)u ¿ 1 '(u)|u|m − 2 |v|m − $1 (t; x); |a(t; x; v; u)| 6 2 ('(u)|u| + |v|)m−1 + $(t; x); where m = (2 + !)=(1 + !) under condition ($1 ) and m = 2 under condition ($2 ), here '(u) is a nonnegative function bounded on R1 . We assume also an additional condition on the kernel K(x; y): (K2 ) if condition ($1 ) is satis8ed, then K(x; y)g(x)g(y) d x dy ¿ 0; ∀g ∈ L∞ ():
Remark 2. In relevant applications the kernel K models nonlocal particle interaction. Positive sign of K as assumed in condition (K2 ), corresponds to repulsive interaction between particles and implies, roughly speaking, global existence of solutions, whereas negative sign models attraction forces and may cause blow up of solutions (cf. [10]). However, under condition ($2 ) assumed in the papers [5,11] turns out to be bounded, so global existence can be proved without condition (K2 ).
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We consider problem (1.1)–(1.4) with f; h such that @f ∈ Lm (0; T; [W 1; m ()]∗ ); f ∈ C([0; T ]; Lp2 ()); @t h(x) ∈ L∞ ()
(2.4) (2.5)
and p2 ¿ n + 2!=(! + 1) in the case of condition ($1 ) and p2 ¿ n under condition ($2 ). Denition 1. A pair of functions (u; v), u; v ∈ L2 (0; T ; W 1; 2 ()), is called a solution of problem (1.1)–(1.4), if the following conditions are satis8ed: (i) the derivative @(u)=@t exists in the sense of distributions,
@u 2 @v 2 (u) + d x dt ¡ ∞; @x @x QT @(u) ∈ L2 (0; T ; [W 1; 2 ()]∗ ); @t (ii) ∀’ ∈ C ∞ (QO T ) and almost every , ∈ (0; T ), Q, = (0; ,) × , , n @(u − v) @’ @(u) (u)bi t; x; u; ; ’ dt + @t @x @xi 0 Q, i=1 (u) ∈ C([0; T ]; L2 ());
+ a(t; x; v; u)’
d x dt = 0;
equality (1.2) is satis8ed for almost all (t; x) ∈ QT ; (iii) ∀’ ∈ C ∞ (QO T ), satisfying ’(T; x) = 0 for x ∈ , T @’ @(u) ; ’ dt + d x dt = 0: [(u) − (h)] @t @t 0 QT
(2.6) (2.7)
(2.8)
(2.9)
Remark 3. Let (u; v) be a solution of problem (1.1)–(1.4). Since the space C ∞ (QO T ) is dense in the weighted space L2 (0; T ; W 1; 2 (; (u)), the integral identity (2.8) holds for all ’ ∈ L2 (0; T ; W 1; 2 ()) such that 2 @’ (u) d x dt ¡ ∞: @x QT
Remark 4. Lemma 1 below guarantees that the right-hand side of equality (1.2) is well-de8ned under our conditions on the functions and f. In what follows, we shall understand as known parameters all numbers from conditions (ii), (iii), (K1 ), norms of functions f; h; $1 , $ in respective spaces, numbers that depend only on ; T; n, the numbers 1 ; 2 ; 3 = max{(u): |u| 6 m0 } and 4 = min{(u) : |u| 6 m0 }, where m0 = h(x)L∞ () + 1: Further we shall denote by cj constants depending only on known parameters.
(2.10)
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Theorem 1. Let conditions (i) – (iii), (K1 ), (K2 ), (1 ), (@), (2.4), (2.5) be satis
@(u − v) 2 (u) dt d x 6 M1 ; ess sup (u(t; x)) d x + (2.11) @x t∈(0;T ) QT where
(u) =
0
u
s(s) ds:
(2.12)
Theorem 2. Let the assumptions of Theorem 1 and condition (2 ) be satis
@u 2 @v 2 (u) + d x dt 6 M2 : (2.13) @x @x QT Theorem 3. Let the assumptions of Theorem 2 be satis
(2.15)
Theorem 5. Let the conditions of Theorem 4 be satis
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H. Gajewski, I.V. Skrypnik / Nonlinear Analysis 56 (2004) 803 – 830
Proposition 1. Let the conditions of Theorem 6 be satis
@u ∈ L∞ (0; T ; L2 ()) ∩ L2 (0; T ; W 1; 2 ()): @t
Remark 5. Proposition 1 and Theorem 4 imply that t → t @(u)=@t ∈ L∞ (0; T ; L2 ()). Consequently, (1.1) can be understood not only in the sense of distributions, but even as an equation in L2 (0; T ; L2 ()). Proofs of Theorems 1–3 are given in Section 3, proofs of Theorems 4–6 are given in Sections 4, 5, 6, respectively. 3. Integral estimates of the solution We start from auxiliary lemmas needed in the proofs of the Theorems 1–6. Let us de8ne operators K0 , K1 for g ∈ L∞ () by @K(x; y) |K(x; y)|g(y) dy; K1 g(x) = K0 g(x) = @x g(y) dy:
(3.1)
Lemma 1. The operators K0 , K1 are well de
for 1 6 p ¡ n;
for 1 6 p 6 ∞:
(3.2) (3.3)
Proof. Firstly, we prove (3.3). For p = 1 and ∞ (3.3) is a simple consequence of (1.9). For 1 ¡ p ¡ ∞ we 8nd for g ∈ Lp () by HPolder’s inequality p @K(x; y) p |K1 g(x)| d x 6 @x |g(y)| dy d x p−1 @K(x; y) @K(x; z) |g(y)|p dy 6 dx @x @x d z 6 –p |g(y)|p dy;
that is (3.3). For proving (3.2) we use the embedding theorem for W 1; 1 () to infer from (1.9) (3.4) ess sup |K(x; y)|n=(n−1) dy 6 c1 : x∈
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Now by HPolder’s inequality we have for 1 6 p ¡ n, g ∈ L∞ (), |K0 g(x)|np=(n−p) d x
6
|K(x; y)|n(p−1)=((n−1)p) · [|K(x; y)|n−p=((n−1)p)
(n−p)=n
× |g(y)| 6
×
] · |g(y)|
|K(x; y)|n=(n−1) dy
p=n
|g(y)| ˜ p d y˜
np=(n−p) dy
dx
(n(p−1))=(n−p) · |K(x; y)| O n=(n−1) · |g(y)| O p d yO
p=(n−p) d x 6 c2
|g(x)|p d x
n=(n−p) :
This inequality implies (3.2) and the proof of Lemma 1 is complete. Lemma 2. Let the assumptions of Theorem 1 be satis
(3.5)
holds for each , ∈ (0; T ) with a constant M5 depending only on known parameters. Proof. Let , ∈ (0; T ) and de8ne for 0 ¡ 2 ¡ T − , , [(u(t + 2; x)) · v(t + 2; x) − (u(t; x))v(t; x)] d x dt: I (2) = 0
(3.6)
By writing I (2) as di#erence of two integrals and changing the integration variable in the 8rst integral we get ,+2 2 I (2) = (u(t; x))v(t; x) d x dt − (u(t; x))v(t; x) d x dt: (3.7) ,
0
On the other hand, we can rewrite I (2) as I (2) = I1 (2) + I2 (2); where I1 (2) = I2 (2) =
, 0
0
,
[(u(t + 2; x)) − (u(t; x))]v(t + 2; x) d x dt; (u(t; x))[v(t + 2; x) − v(t; x)] d x dt:
(3.8)
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H. Gajewski, I.V. Skrypnik / Nonlinear Analysis 56 (2004) 803 – 830
Using (1.2) and setting v1 (t; x) = K(x; y)(u(t; y)) dy, we can rewrite I2 (2) as , [(u(t + 2; x)) − (u(t; x))]v(t; x) d x dt I2 (2) = 0
+
, 0
2
,
−
−
,+2
,
[f(t + 2; x) − f(t; x)]v1 (t; x) d x dt [f(t − 2; x) − f(t; x)]v1 (t; x) d x dt
f(t − 2; x)v1 (t; x) d x dt +
0
2
f(t; x)v1 (t; x) d x dt:
(3.9)
From (3.8)–(3.9), (2.4), (2.7) and Lemma 1 we see that dividing I (2) by 2 and passing to the limit 2 → +0 gives (u(,; x))v(,; x) d x − (h(x))v(0; x) d x
=2
, 0
−
@(u) ;v @t
dt + 2
, 0
f(,; x)v1 (,; x) d x +
@f ; v1 @t
dt
f(0; x)v1 (0; x) d x:
(3.10)
We shall estimate the summands in (3.10). In the case of condition ($1 ) we have by (2.3), (3.2) and condition (K2 ) (u(,; x))v(,; x) d x
=
6
f(,; x)v1 (,; x) −
K(x; y)(u(,; x))(u(,; y)) dy
f(,; x)v1 (,; x) d x 6 c3 (u(,; x)L1 () :
dx (3.11)
An analogous estimate is true under condition ($2 ), because of the boundedness of the function in that case. Further, using Lemma 1 we get , , 1=m @v1 m @f m ; v1 dt 6 c4 @x + v1 d x dt @t 0 0 6 c5 (u)Lm (Q, ) :
(3.12)
Estimating the remaining summands in (3.10) and using (3.11), (3.12), we obtain from (3.10) the desired estimate (3.5) and the proof of Lemma 2 is complete. Proof of Theorem 1. Condition (2.6) and Remark 3 allow us to use the test function ’ = u − v in integral identity (2.8). Then, evaluating the resulting terms by conditions
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(ii), (iii) and Lemma 2, we obtain , @(u − v) 2 @(u) dx (u) ; u dt + @t @x 0 Q, + '(u)|u(t; x)|m d x dt 6 c6 1 + (u(,; x))L1 () Q,
+ (u)Lm (Q, ) +
Q,
[|v(t; x)|m + $1 (t; x) + $1+! (t; x)] d x dt :
We transform the 8rst integral in (3.13) in the following way: , @(u) (u; (,; x)) d x − (h(x)) d x ; u dt = @t 0
(3.13)
(3.14)
with (u) de8ned by (2.9). The proof of equality (3.14) is analogous to the proof of Lemma 1 in [9]. Remarking that condition (2 ) implies c7−1 [(u)]m − c8 6 (u) 6 c7 [(u)]m + c8 ;
(3.15)
using (3.14) and Lemma 1, we obtain from (3.13) @(u − v) 2 m d x dt (u) (u(,; x)) d x + @x Q, + '(u)|u(t; x)|m d x dt 6 c9 1 + m (u(t; x)) d x dt : Q,
Q,
(3.16)
Now estimate (2.11) follows from (3.15), (3.16) and Gronwall’s Lemma and the proof of Theorem 1 is complete. Proof of Theorem 2. The assertion of Theorem 2 follows simply under ($2 ). Indeed, in this case the functions ; are bounded such that (2.4) and Lemma 1 imply @v=@x ∈ C([0; T ], Lp2 ()). Hence (2.13) follows immediately from (2.11). Let us now assume that condition ($1 ) is satis8ed. In this case we de8ne u+ (t; x) = max{u(t; x); 0};
QT± = {(t; x) ∈ QT : ± u(t; x) ¿ 0}:
(3.17)
Theorem 1 implies [1 + u+ (t; x)]!+2 d x dt ess sup t∈(0;T )
+
QT+
@(u − v) 2 d x dt 6 c10 : [1 + u(t; x)] @x !
Thus for proving the desired inequality (2.13), it suMces to show that 2 @v [1 + u+ (t; x)]! d x dt 6 c11 : @x QT
(3.18)
(3.19)
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H. Gajewski, I.V. Skrypnik / Nonlinear Analysis 56 (2004) 803 – 830
De8ne for q ¿ 2=(! + 1), ¿ ! + 2, q ! @v [1 + u+ (t; x)] d x dt; I (q) = @x QT
J( ) =
QT
u+ (t; x) d x dt:
(3.20)
We shall need the following assertions: (1) the estimate I (2=(! + 1)) 6 c12 holds; O 6 c14 with qO = ( O − !)=(! + 1); (2) if for O ¿ ! + 2, J ( O) 6 c13 , then I (q) (3) if for q˜ ∈ [2=(! + 1); 2], I (q) ˜ 6 c15 , then J ( ˜) 6 c16 , with ˜ = q˜ + ! + (q=n)(2 + !). To prove assertion (1) we apply (1.9), Theorem 1, HPolder’s and Young’s inequalities: @K(x; y) 2 ! (2=(!+1))−1 6– [1 + u+ (t; x)] I @y !+1 QT ×|(u(t; y)) − f(t; y)|2=(!+1) dy d x dt 6 c17 –(2=(!+1)) {1 + [u+ (t; x)]2! + |f(t; x)|m } d x dt; QT
(3.21)
where we used also (2.2) and the simple inequality |(u)| 6 c18 (1 + u+ )!+1 ;
u+ = max(u; 0):
(3.22)
Now (3.21), (3.18) and (2.4) imply assertion (1). Assertion (2) follows from the next inequality that is obtained analogously to (3.21). @K(x; y) ! qO q−1 O [1 + u+ (t; x)] I (q) O 6– @y |(u(t; y)) − f(t; y)| dy d x dt QT O q {1 + [u+ (t; x)]!+q(!+1) + |f(t; x)|}!=(!+1)+qO d x dt 6 c20 : (3.23) 6 c19 – QT
Assertion (3) follows by HPolder’s inequality and Sobolev’s embedding theorem. Indeed, we get from (3.18) q=n T ˜ q˜ 2+! [1 + u+ (t; x)] d x J q˜ + ! + (2 + !) 6 n 0 (n−q)=n ˜ (q+!)n=(n− ˜ q) ˜ × [1 + u+ (t; x)] dx dt
6 c21
QT+
q˜ @u [1 + u(t; x)]! d x dt @x
+ c21
QT
2+!
[1 + u+ (t; x)]
(!+q)=(!+2) ˜ d x dt
:
(3.24)
H. Gajewski, I.V. Skrypnik / Nonlinear Analysis 56 (2004) 803 – 830
Inequalities I (q) ˜ 6 c15 and (3.18) imply q˜ ! @u [1 + u(t; x)] d x dt 6 c22 ; + @x QT
815
(3.25)
we obtain assertion (3) from (3.24), (3.25) and (3.18). Let us de8ne sequences {qi }, { i }, i = 1; : : : ; N , such that q1 =
2 ; !+1
i
qN −1 ¡ 2;
= qi + ! +
qi (2 + !); n
qi+1 =
i −! ; !+1
qN ¿ 2:
(3.26)
This de8nition is justi8ed by (2.2) and qi 2 [2 − !(n − 1)] ¿ 0: [2 − !(n − 1)] ¿ n(! + 1)2 n(! + 1)
qi+1 − qi =
Now, using the assertions (1) – (3), we get by iteration that I (qN ) 6 c23 and hence (3.19). This ends the proof of Theorem 2. Proof of Theorem 3. Analogously as in the proof of Theorem 2, we can restrict us to the case of condition ($1 ). We test integral identity (2.8) with ’(t; x) = [(uk (t; x)) − 0 ]+ {1 + [(uk (t; x)) − 0 ]2 }r ;
(3.27)
where uk (t; x) = min {u(x; t); k}, k ¿ m0 , m0 is given by (2.10), 0 = (m0 ) and r ∈ (− 1=2; ∞) is an arbitrary number. Analogously to Lemma 1 in [9] we have , @(u) (r) (u(,; x)) d x; (3.28) ; ’ dt = @t 0 where (r) (u) =
0
u
(s)[(sk ) − 0 ]+ {1 + [(sk ) − 0 ]2 }r ds;
sk = min {s; k} (3.29)
and (r) (u) ¿
1 {1 + [(uk ) − 0 ]2 }r+1 − 1 2(r + 1)
for u ¿ m0 :
(3.30)
We write the derivative of ’ in the form @’ @u = (u)8(r) (uk ) 9(m0 ¡ u ¡ k); @xi @xi
(3.31)
where 9(m0 ¡ u ¡ k) is the characteristic function of the set {(t; x) ∈ QT : m0 ¡ u(t; x) ¡ k} and the function 8(r) (u) satis8es c23 r{1 + [(u) − 0 ]2 }r 6 8(r) (u) 6 c24 (r + 1){1 + [(u) − 0 ]2 }r
(3.32)
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H. Gajewski, I.V. Skrypnik / Nonlinear Analysis 56 (2004) 803 – 830
with r = min (1 + 2r; 1). Using (3.28)–(3.32) and conditions (ii), (iii), we obtain {1 + [(uk (,; x)) − 0 ]2 }r+1 9(m0 ¡ u) d x
+
2 @u 2 (u){1 + [(uk ) − 0 ]2 }r · 9(m0 ¡ u ¡ k) d x dt @x 0
2 2 , @v r+1 6 c24 2 (u){1 + [(uk ) − 0 ]2 }r r @x 0
,
×9(m0 ¡ u ¡ k) d x dt +
r+1 [|u|m−1 + |v|m−1 + $(t; x)] 1+ r Q,
× {1 + [(uk ) − 0 ]2+ }r+1=2 9(m0 ¡ u) d x dt
:
(3.33)
We introduce the notations {u ¿ 1} = {(t; x) ∈ QT : u(t; x) ¿ 1} and 2 @u I ∗ (q) = ess sup q (u+ (t; x)) d x + 2 (u)q−2 (u) d x dt; @x t∈(0;T ) {u¿1} 2+! [1 + u+ (t; x)] d x dt; q ¿ J ∗( ) = ; ¿ ! + 2: (3.34) 1 +! QT We shall need the following assertions: (1) I ∗ ((2 + !)=(1 + !)) 6 c25 ; (2) if J ∗ ( O) 6 c26 , O ¿ ! + 2, then I ∗ (q) O 6 c27 , qO = min {( O − 2!)=(! + 1); p2 − 2!= (! + 1); O=(1 (! + 1)) + 1}; (3) if I ∗ (q) ˜ 6 c28 for q˜ ¿ (2+!)=(1+!), then J ∗ ( ˜) 6 c29 for ˜ =(1=n)q(n+2)(1+!). ˜ Remarking that 2 (u)q−2 (u) 6 c30 (u) for u ¿ 1, q = (2 + !)=(1 + !), we obtain assertion (1) immediately from Theorems 1 and 2. To prove the assertion (2) we start estimating the 8rst integral on the right-hand side of (3.33) with r 6 min {( O − 2!)=(2(! + 1)) − 1, (p2 =2) − (!=(! + 1)) − 1}. Analogously to inequality (3.21) we have 2 , 2 r @v 2 (u){1 + [(uk ) − 0 ] } 9(m0 ¡ u ¡ k) d x dt @x 0 2 2!+2r(1+!) @v [1 + uk (t; x)] 6 c31 @x ) d x dt Q,+ @K(x; y) 2!+2r(1+!) 6 c31 – [1 + uk (t; x)] @y Q,+ ×|(u(t; y)) − f(t; y)|2 dy d x dt
H. Gajewski, I.V. Skrypnik / Nonlinear Analysis 56 (2004) 803 – 830
6 c32
QT
6 c33
817
{1 + |(u(t; x))| + |f(t; x)|}2!=(!+1)+2r+2 d x dt
{1 + [u+ (t; x)]!+1 + |f(t; x)|}2!=(!+1)+2r+2 d x dt 6 c34 :
QT
(3.35)
Let us now estimate the last integral in (3.33). Using Theorem 1, Lemma 1, HPolder’s inequality and supposing r such that m m 6 O; (2r + 1)(! + 1)p1 6 O; 6 p2 ; 2(! + 1) r + 2 r+ 2 2 we obtain Q,+
[|u|m−1 + |v|m−1 + $(t; x)]{1 + [(uk ) − 0 ]2+ }r+(1=2) d x dt
6 c34
Q,+
+
+
(|(u)| + |f(t; x)|)2(r+(m=2)) d x dt
Q,
[1 + uk ]2(!+1)(r+(m=2)) dt d x
[1 + uk ](2r+1)(!+1)p1 d x dt
Q,+
1=p1
6 c35 :
(3.36)
From (3.35) and (3.36) we see that the left-hand side in (3.33) is bounded by some constant depending only on known parameters and independent of k; r, provided J ∗ ( O) 6 c26 and r is de8ned by
O − 2! O 2! 1 r = min + 1 − 1: (3.37) ; p2 − ; 2 !+1 ! + 1 p1 (! + 1) O 6 c27 . That is So we are able to pass to the limit k → +∞ in (3.33) to obtain I ∗ (q) assertion (2). Assertion (3) follows from HPolder’s inequality and Sobolev’s embedding theorem analogously to inequality (3.24). Now we de8ne numbers {qi }, { i }, i = 1; : : : ; N , such that q1 =
2+! ; 1+!
i
=
1 qi (n + 2)(1 + !); n
qi+1 = min
− 2! 2! i +1 ; ; p2 − ; 1+! 1 + ! p1 (! + 1)
qN −1 ¡ p2 −
2! ; !+1
i
qN = p2 −
2! : !+1
(3.38)
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H. Gajewski, I.V. Skrypnik / Nonlinear Analysis 56 (2004) 803 – 830
This de8nition is justi8ed, since {qi } is increasing by − 2! 2qi 2! 2 − qi = − ¿ [! + 2 − n!] ¿ 0; !+1 n !+1 n(! + 1) p1 − 1 n + 2 i = q · + 1 − q − 1 + 1 ¿ 1: i i p1 (! + 1) p1 n i
(3.39)
Note also that N ¿ (n + 2)(1 + !). So assertions (1) – (3) imply I ∗ (qi ) 6 c36 , J ∗ ( i ) 6 c37 for i = 1; : : : ; N . By I ∗ (qN ) 6 c36 , J ∗ ( N ) 6 c37 we have [(u(t; x))]p2 −2!=(!+1) d x 6 c36 ; ess sup t∈(0;T )
QT
|(u(t; x))|
N =(1+!)
d x dt 6 c37 :
(3.40)
Hence conditions (2.4), (’) and Lemma 1 imply (2.14) with p3 = min {p2 − 2!= (! + 1); N =(! + 1) − 2} and the proof of Theorem 3 is complete.
4. Boundedness of the function u Firstly we want to estimate u(t; x) from above under condition ($1 ). Lemma 3. Let the conditions of Theorem 4 and ($1 ) be satis
(4.1)
Proof. We apply (3.33) and estimate the integrals on the right-hand side by HPolder’s inequality. Using the properties of the function $, (2.14) and (3.40), we get 2 @v 2 (u){1 + [(uk ) − 0 ]2 } 9(m0 ¡ u ¡ k) d x dt @x Q, + [|u|m−1 + |v|m−1 + $(t; x)]{1 + [(uk ) − 0 ]2 }r+1=2 9(m0 ¡ u) d x dt Q,
6 c39
Q,
2 (r+1)pO
{1 + [(uk ) − 0 ] }
1=pO 9(m0 ¡ u) d x dt
(4.2)
with pO ¡ (n + 2)=n depending only on known parameters. Inequalities (3.33), (4.2) imply for r ¿ 1 {1 + [(uk (,; x)) − 0 ]2 }r+1 9(m0 ¡ u) d x
H. Gajewski, I.V. Skrypnik / Nonlinear Analysis 56 (2004) 803 – 830
2 @u 2 (u){1 + [(u) − 0 ]2 }r 9(m0 ¡ u ¡ k) d x dt @x Q, 1=pO 2 (r+1)pO 2 6 c40 r 1 + {1 + [(uk ) − 0 ] } 9(m0 ¡ u) d x dt :
819
+
Q,
(4.3)
Last estimate, HPolder’s inequality and Sobolev’s embedding inequalities yield for r ¿ 1 {1 + [(uk ) − 0 ]2 }(r+1)(n+2)=n · 9(m0 ¡ u) d x dt QT
6 c41 · ×
T
0
2 r+1
{1 + [(uk ) − 0 ] }
2=n · 9(m0 ¡ u) d x
{1 + [(uk ) − 0 ]2 }(r+1)n=(n−2) d x
6 c42 r 2 ess sup
0¡t¡T
(n−2)=n dt
{1 + [(uk (t; x)) − 0 ]2 }r+1 9(m0 ¡ u) d x
2=n
2 @u 2 (u){1 + [(u) − 0 ]2 }r 9(m0 ¡ u ¡ k) @x QT +{1 + [(uk ) − 0 ]2+ }r+1 d x dt
×
6 c43 r
4+4=n
QT
{1 + [(uk ) −
0 ]2+ }(r+1)pO
(1+(2=n))1=pO d x dt
:
(4.4)
Inequalities (4.4) and (3.40) justify the application of Moser’s iteration process to verify (4.1) and the proof of Lemma 3 is complete. For arbitrary k ∈ R and functions w on QT we de8ne w(k) = w(k) (t; x) = max {w(t; x); k};
w− = w− (t; x) = min{w(t; x); 0}:
(4.5)
Lemma 4. Let the conditions of Theorem 4 be satis
(4.7)
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H. Gajewski, I.V. Skrypnik / Nonlinear Analysis 56 (2004) 803 – 830
Proof. We test the integral identity (2.8) with 1 ’= [(u(k) ) − (−m0 )]− · |u(k) + m0 |r ; k ¡ − m0 ; r ¿ 0: (u(k) ) Then, analogously to the proof of inequality (4.32) in [7], we obtain |[u(k) (,; x) + m0 ]− |r+1 d x
+
Q,
2 @u |u + m0 | 9(k ¡ u ¡ − m0 ) d x dt @x r
2 2 2
@u @v r @v 6 c44 (r + 1) |u + m0 | + + @x @x @x Q, 2
× 9(k ¡ u ¡ − m0 ) + [1 + $(t; x)] · |[u(k) + m0 ]− |r
d x dt:
(4.8)
Using Lemma 4 we have from (4.8) 2 @u |u + m0 |r 9(k ¡ u ¡ − m0 ) d x dt |[u(k) (,; x) + m0 ]− |r+1 d x + @x Q, 2 (k) r 6 c45 (r + 1) |[u + m0 ]− | $(t; ˜ x) d x dt + 1 ; (4.9) Q,
where $(t; ˜ x)=$(t; x)+|@v=@x|2 +1. The condition on $ and Theorem 3 imply $˜ ∈ Lp˜ (QT ) with p˜ ¿ (n + 2)=2. Inequality (4.9) allows to apply Moser’s iteration process for proving |[u(k) (t; x) + m0 ]− | 6 c46 : This implies (4.7) with M8 = m0 + c46 and Lemma 5 is proved. Proof Theorem 4. The assertion of Theorem 4 follows immediately from the Lemmas 3, 5 if condition ($1 ) is satis8ed. In the case of condition ($2 ) Lemma 5 yields a lower bound of u(t; x). The existence of an upper bound in that case can be analogously shown. The proof of Theorem 4 is complete. 5. Proof of the existence theorem Firstly we shall assume that condition ($1 ) is satis8ed. In this case we regularize the problem (1.1)–(1.4) by replacing ; a; by ∗ ; a∗ ; ∗ in the following way: Let M4 be the constant from Theorem 4 and (t; x) ∈ QT , v ∈ R1 , then ∗ (u) = (u); a∗ (t; x; v; u) = a(t; x; v; u); ∗ (u) = (u)
if u 6 M4 ;
(5.1)
∗ (u) = (M4 )eM4 −u ; a∗ (t; x; v; u) = a(t; x; v; M4 )eM4 −u ; ∗ (u) = (M4 ) + (M4 )[1 − eM4 −u ]
if u ¿ M4 :
(5.2)
H. Gajewski, I.V. Skrypnik / Nonlinear Analysis 56 (2004) 803 – 830
We consider the regularized problem in QT , i.e., n @(u − v) @ @∗ (u) + a∗ (t; x; v; u); ∗ (u)bi t; x; − @x @t @x i c=i v(t; x) = − K(x; y)[∗ (u(t; y)) − f(t; y)] dy; n
bi
i=i
@(u − v) t; x; @x
u(0; x) = h(x);
821
(5.3) (5.4)
cos (; xi ) = 0
(t; x) ∈ (0; T ) × @;
x ∈ :
(5.5) (5.6)
This problem satis8es all conditions of Section 2 with the same known parameters as problem (1.1)–(1.4). Therefore each solution (u; v) of problem (5.3)–(5.6) satis8es a priori estimate (2.15). So from (5.1) we see that a solution (u; v) of problem (5.3)–(5.6) is automatically solution of problem (1.1)–(1.4). Therefore it is suMcient to establish the existence of a solution of problem (5.3)–(5.6) in order to prove the Theorem 5. Let X (k); k ∈ [0; 1], be the Banach space of functions such that |u(t + 2; x) − u(t; x)|2 d x dt ¡ ∞: u2X (k) = u2L2 (0; T ;W 1; 2 ()) + sup 2k 0¡2¡T=2 QT −2 To study the solvability of problem (5.3)–(5.6) we introduce the operator A : X (1=2) → X (1=2) transforming a function g ∈ X (1=2) into the solution U = Ag of the following problem in QT : n @(U − G) @∗ (U ) @ + a∗ (t; x; G; U ) = 0; ∗ (U )bi t; x; (5.7) − @t @xi @x i=1 G(t; x) = − K(x; y)[∗ (g(t; y)) − f(t; y)] dy; (5.8) n i=1
bi
@(U − G) t; x; @x
U (0; x) = h(x);
cos(; xi ) = 0;
x ∈ :
(t; x) ∈ (0; T ) × @;
(5.9) (5.10)
Taking into account the boundedness of the function ∗ , assumptions (2.4), (@) and Lemma 1 we have @G(t; x) p2 (5.11) ess sup @x d x + ess sup |G(t; x)| 6 c47 ; t∈(0;T )
(t;x)∈QT
with a constant c47 depending only on known parameters and independent of g. In order to guaranty the unique solvability of problem (5.7), (5.9), and (5.10) for given function G satisfying (5.11), the Theorems 3, 4 in [9] can be adapted. Indeed, the functions @G(t; x) ; i = 1; : : : ; n b∗i (t; x; &) = bi t; x; & − @x
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H. Gajewski, I.V. Skrypnik / Nonlinear Analysis 56 (2004) 803 – 830
satisfy the inequalities n [b∗i (t; x; & ) − b∗i (t; x; & )](&i − &i ) ¿ |& − & |2 ;
(5.12)
i=1
|b∗i (t; x; &)| 6 2 |&| + ?(t; x)
(5.13)
with ?(t; x) = 2 (1 + |@G=@x|) ∈ L∞ (0; T ; Lp2 ()), which essentially coincide with conditions (ii) ensuring in [9] existence and uniqueness in the case of Dirichlet boundary conditions. But it is simple to check that the Theorems 3 and 4 in [9] are also true for Neumann boundary conditions. Estimate (5.11) and adaptations of the Theorems 1 and 2 from [9] imply @U (t; x) 2 ess sup {|U (t; x)|: (t; x) ∈ QT } 6 M9 ; (5.14) @x d x dt 6 M9 ; QT where U (t; x) is the solution of problem (5.7), (5.9), (5.10) and M9 is a constant depending only on known parameters and independent of g. Using estimates (5.14) and (5.11) we can show analogously to [14] that |U (t + 2; x) − U (t; x)|2 sup d x dt 6 M10 ; (5.15) 2 0¡2¡T=2 QT −2 with a constant M10 depending only on known parameters and independent of g. So the solution U of problem (5.7), (5.9), (5.10) belongs to the space X (1) and therefore the operator A : X (1=2) → X (1=2) is well de8ned. From the de8nition of this operator we see immediately that the solvability of problem (5.3)–(5.6) is equivalent to the existence of a 8xed point of the operator A Ag = g; g ∈ X 12 : (5.16) We shall prove the existence of a solution of (5.16) by using the Leray–Schauder principle. The Leray–Schauder degree theory implies (cf. [14,17]) that for the solvability of Eq. (5.16) it is suMcient to establish following statements: (1) there exists a family {A@ }, @ ∈ [0; 1], of operators A@ : X (1=2) → X (1=2) such that A1 = A, A0 = 0, A@ is completely continuous for each @ ∈ [0; 1] and {A@ } satis8es following continuity condition: for arbitrary sequences {@j }, {uj } such that @j → @0 , uj → u0 we have A@j uj → A@0 u0 , where → denotes strong convergence in X (1=2). (2) there exists a positive number R such that A@ g = g
for @ ∈ [0; 1];
g
1 X(2)
= R:
(5.17)
We de8ne A@ g = U@ , where U@ is the solution of the problem n @∗ (U@ ) @ @(U@ − @G) ∗ − (U@ )bi t; x; @t @xi @x i=1
+ a∗ (t; x; G; U@ ) − (1 − @)a∗ (t; x; G; 0) = 0; (t; x) ∈ QT ;
(5.18)
H. Gajewski, I.V. Skrypnik / Nonlinear Analysis 56 (2004) 803 – 830 n
bi
i=1
@(U@ − @G) t; x; @x
U@ (0; x) = @h(x);
823
cos (; xi ) = 0;
(t; x) ∈ (0; T ) × @;
x ∈ :
(5.19) (5.20)
The function G(t; x) in (5.18) – (5.20) is given by (5.8). The unique solvability of this problem can be seen as that of (5.7), (5.9), (5.10). Hence the operator A@ : X (1=2) → X (1=2) is well de8ned. We shall check 8rstly statement (2) formulated above. Let us assume that @; g are such that @ ∈ [0; 1], g ∈ X (1=2) and A@ g = g. Then from (5.18)–(5.20), (5.8) we see that the pair (U; G) is a solution of a nonlocal nonlinear problem being analogous to problem (1.1)–(1.4). Consequently, by Theorem 2 there exists a constant M11 depending only on known parameters and independent of @ ∈ [0; 1] such that U@ L2 (0; T ;W 1; 2 ()) 6 M11 . From the corresponding inequality (5.15) we have U@ X (1=2) 6 M12 with a constant M12 depending only on known parameters and independent of g(t; x); @. Since the equality A@ g=g implies gX (1=2) 6 M12 , the desired relation (5.17) is ful8lled for R = M12 + 1. Now we shall check statement (1) formulated above. The equalities A1 = A and A0 = 0 hold because of the unique solvability of problem (5.7), (5.9), (5.10). Thus it remains to prove compactness and continuity of the operator A. To this aim we prove the following lemma. Lemma 6. Assume that the conditions of Theorem 4 are satis
We put the test functions ’i given by ’1 =
1 [∗ (U1 ) − ∗ (U2 )]; ∗ (U1 )
’2 = U1 − U2 ;
into the integral identities corresponding to Ui . Taking the di#erence of the two resulting equalities, we get ∗ , ∗ @ (U1 ) @ (U2 ) 1 dt ; ∗ [∗ (U1 ) − ∗ (U2 )] − ; U1 − U2 @t (U1 ) @t 0 n @(U1 − G1 ) @ 1 ∗ ∗ + ( ∗ (U1 )bi t; x; (U ) − (U )) 1 2 @x @xi ∗ (U1 ) Q, i=1
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H. Gajewski, I.V. Skrypnik / Nonlinear Analysis 56 (2004) 803 – 830
∗
− (U2 )bi
@(U2 − G2 ) t; x; @x
@ [U1 − U2 ] @xi
d x dt
∗ (U1 ) − ∗ (U2 ) ∗ (U1 ) Q, − a∗ (t; x; G2 ; U2 )(U1 − U2 ) d x dt = 0: a∗ (t; x; G1 ; U1 )
+
(5.22)
We transform the 8rst integral in (5.22) analogously to Lemma 2 in [9] to obtain ∗ , ∗ @ (U1 ) @ (U2 ) 1 ∗ ∗ dt ; ∗ [ (U1 ) − (U2 )] − ; U1 − U2 @t (U1 ) @t 0 U1 (,;x) ∗ = [U1 (,; x) − s] (s) ds d x ¿ c48 |U1 (,; x) − U2 (,; x)|2 d x: (5.23)
U2 (,;x)
To estimate the second integral in (5.22), we note that by condition (1 ) U1 ∗ ( ) (s) ∗ (∗ ) (U1 ) ∗ ∗ (s) ds − [ (U1 ) − (U2 )] ¿ − ∗ (s) ∗ (U1 ) U2 = ∗ (U2 ) − ∗ (U1 ); such that n
∗
(U1 )bi
i=1
¿
@(U1 − G1 ) t; x; @x
@ @xi
1 (∗ (U1 ) − ∗ (U2 )) ∗ (U1 )
(5.24)
n @U1 ∗ @(U1 − G1 ) @G1 − bi t; x; − bi t; x; [ (U2 ) − ∗ (U1 )] @x @x @xi i=1
n @(U1 − G1 ) @U1 @U2 ∗ ∗ (U1 ) bi t; x; − (U2 ) + @x @xi @xi i=1
n @G1 @U1 (∗ ) (U1 ) ∗ − bi t; x; − · ∗ [ (U1 ) − ∗ (U2 )]: @x @xi (U1 )
(5.25)
i=1
Since the properties of the function ensure that ∗ ∗ (U1 ) − ∗ (U2 ) − ( ) (U1 ) · [∗ (U1 ) − ∗ (U2 )] 6 c49 |U1 − U2 |; ∗ (U1 ) we get from (5.11), (5.25), (5.26) and condition (ii) n @(U1 − G1 ) @ 1 ∗ ∗ ∗ (U1 )bi t; x; ( (U1 ) − (U2 )) @x @xi ∗ (U1 ) i=1
∗
− (U2 )bi
@(U2 − G2 ) t; x; @x
@ (U1 − U2 ) @xi
(5.26)
H. Gajewski, I.V. Skrypnik / Nonlinear Analysis 56 (2004) 803 – 830
825
2 @(U1 − U2 ) 2 − c51 @(G1 − G2 ) + 1 + @G1 @U1 ¿ c50 @x @x @x @x × |U1 − U2 | :
(5.27)
Using condition (iii) and (5.14), we can estimate the last integral in (5.22) ∗ ∗ ∗ a (t; x; G1 ; U1 ) (U1 ) − (U2 ) − a∗ (t; x; G2 ; U2 )(U1 − U2 ) ∗ (U1 ) 6 c52 |U1 − U2 | [1 + $(t; x)]:
(5.28)
Finally, from (5.22), (5.23), (5.27) and (5.28) we see that @(U1 − U2 ) 2 2 d x dt |U1 (,; x) − U2 (,; x)| d x + @x Q,
@(G1 − G2 ) 2 @G1 @U1 6 c53 + 1 + @x @x · |U1 − U2 | @x Q, +[1 + $(t; x)] |U1 − U2 |
d x dt:
(5.29)
Now we are ready to return to the study of properties of the operator A. We begin with the compactness. Let {gj } be a bounded sequence in X (1=2). Then by the compactness of the embedding X (1=2) ⊂ L2 (QT ) we can assume that {gj } converges strongly in L2 (QT ) to some function g0 . This and Lemma 1 imply the strong convergence of @Gj =@x to @=@xG0 in [L2 (QT )]n , where Gj is de8ned analogously to (5.21). Using (5.14), (5.15) with Uj = Agj , we can assume that Uj converges to some U0 ∈ X (1=2) weakly in L2 (0; T ; W 1; 2 ()) and strongly in Lq (QT ) for an arbitrary q ¡ ∞. In order to prove strong convergence of {Uj } in L2 (0; T ; W 1; 2 ()), we use (5.29) with U1 = Uj , U2 = Ui , G1 = Gj , G2 = Gi and we obtain @(Uj − Ui ) 2 d x dt |Uj (,; x) − Ui (,; x)|2 d x + ess sup @x ,∈(0;T ) QT
@(Gj − Gi ) 2 @Gj @Uj 6 c54 + 1 + @x · @x · |Uj − Ui | @x QT + [1 + $(t; x)]|Ui − Uj |
d x dt:
(5.30)
Using already known convergence properties of the sequences {Uj }, {Gj } and (5.11), we see that the right-hand side of (5.30) tends to zero as j; i → ∞. That means compactness of the sequence {Uj } in L2 (0; T ; W 1; 2 ()). The compactness of this sequence in X (1=2) follows now from (5.30) and (5.15) with Uj , Ui . So we have established the compactness of the operator A. Now we shall check its continuity. Let {gj } be a sequence converging strongly in X (1=2) to g0 . Lemma 1 implies that
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(@=@x)Gj → (@=@x)G0 in [L2 (QT )]n . Using the compactness of A we can assume that {Uj = Agj } converges strongly in X (1=2) to some UO 0 ∈ X (1=2). We have to show UO 0 = Ag0 . From the integral identity for Uj , ∗ n @(Uj − Gj ) @’ @ (Uj ) ; ’ dt + ∗ (Uj )bi t; x; @t @x @xi 0 Q, i=1 + a∗ (t; x; Gj ; Uj )’
d x dt = 0;
’ ∈ L2 (0; T ; W 1; 2 ())
(5.31)
we obtain the boundedness of the sequence {(@=@t)∗ (Uj )} in L2 (0; T ; [W 1; 2 ()]∗ ). Therefore, we can assume that ∗ (Uj ) converges weakly in H 1 (0; T ; [W 1; 2 ()]∗ ) to some functional h0 . Using the strong convergence of {Uj } to UO 0 in L2 (QT ), it is simple to see that h0 = ∗ (UO 0 ). Now we are able to pass to the limit j → ∞ in (5.31) to get , ∗ O n @(UO 0 − G0 ) @’ @ (U 0 ) ∗ O ; ’ dt + (U 0 )bi t; x; @t @x @xi 0 Q, i=1 + a∗ (t; x; G0 ; UO 0 )’
d x dt = 0;
UO 0 (0; x) = h(x); x ∈ :
’ ∈ L2 (0; T ; W 1; 2 ()); (5.32)
Adapting the uniqueness result of Theorem 4 from [9], we obtain from (5.32) UO 0 =Ag0 and this ends the proof of Lemma 6. End of the proof of Theorem 5. We have had reduced the solvability of problem (1.1)– (1.4) to that of Eq. (5.16). The solvability of the last equation follows via Leray– Schauders’s principle from the above formulated statements (1), (2), which are consequences of Lemma 6. Therefore the proof of Theorem 5 is complete provided condition ($1 ) is satis8ed. In the case of condition ($2 ) the same arguments can be used. But it is not necessary to pass to the regularized problem (5.3)–(5.5) in that case.
6. Proof of the Uniqueness Theorem Assume by contradiction the existence of two solutions (uj ; vj ), j = 1; 2; of problem (1.1)–(1.4) in the sense of De8nition 1. We shall show that u1 (t; x) = u2 (t; x), v1 (t; x) = v2 (t; x). By Theorems 2–4 we have @uj @vj + 6 M13 (6.1) uj L∞ (QT ) + vj L∞ (QT ) + @x L2 (QT ) @x L2 (QT ) with a constant M13 depending only on known parameters. Let us now prove two auxiliary estimates.
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First auxiliary estimate: We have for almost all , ∈ (0; T ) @(u1 − u2 ) 2 2 d x dt |u1 (,; x) − u2 (,; x)| + @x Q, @(v1 − v2 ) 2 + |v1 − v2 |2 6 c55 @x Q, @v1 + @x
@u1 + 1 + $(t; x) |u1 − u2 |2 d x dt: @x
(6.2)
We shall obtain this estimate from equality (5.22) with (ui ), (ui ), a(t; x; vi , ui ), ui , vi instead of ∗ (Ui ), ∗ (Ui ), a∗ (t; x; Gi , Ui ), Ui , Gi respectively. Indeed, using (6.1) and the local Lipschitz conditions for resp. for a(t; x; ·; u), we get (u1 ) − (u2 ) − (u1 ) [(u1 ) − (u2 )] 6 c56 |u1 − u2 |2 (6.3) (u1 ) and
a(t; x; v1 ; u2 ) (u1 ) − (u2 ) − a(t; x; v2 ; u2 )(u1 − u2 ) (u1 ) (u1 ) − (u2 ) 6 |a(t; x; v1 ; u2 )| · − (u1 − u2 ) (u1 ) +|a(t; x; v1 ; u2 ) − a(t; x; v2 ; u2 )| · |u1 − u2 | 6 c57 {[1 + $(t; x)] |u1 − u2 |2 + |v1 − v2 |2 }:
(6.4)
Now (6.2) follows by using (6.3) and (6.4) in the same way as (5.29). Second auxiliary inequality: We have for almost all , ∈ (0; T )
2 @u2 2 2 2 @u1 d x dt |u1 − u2 | |u1 (,; x) − u2 (,; x)| d x + + @x @x Q, @(u1 − u2 ) 2 @(v1 − v2 ) 2 + 6 c58 @x @x Q,
@v1 2 @v2 2 2 2 |u1 − u2 | + [1 + $(t; x)] |u1 − u2 | d x dt: + + @x @x
(6.5)
The proof of inequality (6.5) coincides with that of inequality (6.8) in [7]. That proof is based on testing integral identity (2.8) for u = uj , v = vj with the test functions ’j given by 1 ’1 = [exp(N(u1 )) − exp(N(u2 ))]+ ; ’2 = N [u1 − u2 ]+ exp(N(u2 ))]; (u1 ) where N = max {|1 − 2 (s)|=2 (s): |s| 6 M13 } and M13 is the constant from (6.1). Remark that this proof is independent of the equation for the function v.
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We shall use also the estimate @(v1 − v2 ) 2 2 dx + |u1 − u2 |2 d x |v − v | d x 6 c 1 2 59 @x
(6.6)
following from Lemma 1 and (6.1). Now we can turn to the proof of Theorem 6. Applying Cauchy’s inequality to the term with |@v1 =@x| in (6.2), we obtain from (6.2), (6.5), (6.6) @(u1 − u2 ) 2 2 d x dt |u1 (,; x) − u2 (,; x)| + @x Q, @v1 2 @v2 2 |u1 − u2 |2 d x dt: 6 c60 1 + $(t; x) + + (6.7) @x @x Q, From condition (iii) and Theorem 3 we have @v1 2 @v2 2 p˜ + 1 + $(t; x) + @x ∈ L (QT ); @x
p˜ = min
p1 ;
p3 + 2 2
¿
n+2 : 2
Estimating the integral on the right-hand side of (6.7) by HPolder’s inequality, we get @v1 2 @v2 2 |u1 − u2 |2 d x dt 1 + $(t; x) + + @x @x Q, 6 c61
2p˜
|u1 − u2 |
Q,
1=p˜ d x dt
:
(6.8)
Applying HPolder’s and Young’s inequalities and the embedding V 2 (Q, ) → L2(n+2)=n (Q, ), where @u(t; x) 2 2 2 uV 2 (Q, ) := ess sup u (t; x) d x + @x dt d x t∈(0;,)
Q,
(cf. [14]), we can estimate the last integral in (6.8) as follows: 1=p˜ 2p˜ |u1 − u2 | d x dt Q,
@(u1 − u2 ) 2 d x dt |u1 (@; x) − u2 (@; x)| d x + 6 ' sup @x 0¡@¡, Q, |u1 − u2 |2 d x dt + c62 '−(n+2)(p˜ −1)=(n+2−np˜ )
2
Q,
(6.9)
with an arbitrary positive number '. With a suitable ' inequalities (6.7)–(6.9) imply |u1 (t; x) − u2 (t; x)|2 d x dt |u1 (,; x) − u2 (,; x)|2 d x 6 c63 (6.10)
Q,
for all , ∈ (0; T ). Gronwall’s lemma and the last estimate yield u1 = u2 . By (6.6) this implies v1 = v2 and the proof of Theorem 6 is complete.
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Proof of Proposition 1. With the solution u of (1.1)–(1.4) we de8ne u1 (t; x) = u(t; x)
u2 (t; x) = u(t + 2; x);
2 ∈ (0; T − t)
and test the integral identity (2.8) with the functions ’i , i = 1; 2, given by t2 ’1 (t; x) = ∗ [∗ (u1 (t; x)) − ∗ (u2 (t; x))]; (u1 (t; x)) ’2 (t; x) = t 2 (u1 (t; x) − u2 (t; x)): Then, arguing essentially as in the proof of (6.7) and (6.10), we obtain @(u1 − u2 ) 2 d x dt ,2 |u1 (,; x) − u2 (,; x)|2 d x + t 2 @x Q, t|u1 − u2 |2 d x dt: 6 c64 Q,
(6.11)
Then, using the interpolation inequality c v2L2 () 6 'Sv2L2 () + v2[W 1; 2 ()]∗ : ' with v(x) = u1 (t; x) − u2 (t; x), ' = t' we can replace inequality (6.11) by 2 2 2 2 @ |u1 (,; x) − u2 (,; x)| d x + t (u1 − u2 ) d x dt , @x
6 c65
0
Q,
,
u1 (t; ·) − u2 (t; ·)2[W 1; 2 ()]∗ dt:
Since @u=@t ∈ L2 (0; T ; [W 1; 2 ()]∗ ), the di#erence quotient (1=2)[u(t + 2; x) − u(t; x)], is bounded in L2 (0; T ; [W 1; 2 ()]∗ ) uniformly with respect to 2. The last inequality now implies uniform boundedness of t [u(t + 2; x) − u(t; x)] 2 in L∞ (0; T ; L2 ()) ∩ L2 (0; T ; [W 1; 2 ()]∗ ) that implies the assertion of Proposition 1. References [1] G. Albinus, H. Gajewski, R. HPunlich, Thermodynamic design of energy models of semiconductor devices, Nonlinearity 15 (2002) 367–383. [2] H.W. Alt, S. Luckhaus, Quasilinear elliptic–parabolic di#erential equations, Math. Z. 183 (1983) 311–341. [3] Ph. Benilan, P. Wittbold, On mild and weak solutions of elliptic–parabolic systems, Adv. Di#erential Equations 1 (1996) 1053–1073. [4] C.M. Elliot, H. Garcke, On the Cahn–Hilliard equation with degenerate mobility, SIAM J. Math. Anal. 27 (1996) 404–423. [5] H. Gajewski, On a nonlocal model of nonisothermal phase separation, Adv. Math. Sci. Appl. 12 (2002) 569–586. [6] H. Gajewski, K. GrPoger, Reaction–di#usion processes of electrically charged species, Math. Nachr. 177 (1996) 109–130.
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[7] H. Gajewski, I.V. Skrypnik, On the uniqueness of solutions for nonlinear elliptic–parabolic equations, J. Evolution Equations 3 (2003) 247–281. [8] H. Gajewski, I.V. Skrypnik, To the uniqueness problem for nonlinear elliptic equations, Nonlinear Anal. 52 (2003) 291–304. [9] H. Gajewski, I.V. Skrypnik, On the uniqueness of solutions for nonlinear parabolic equations, Discrete Continuous Dyn. Systems 10 (2004) 215–236. [10] H. Gajewski, K. Zacharias, Global behavior of a reaction–di#usion system modelling chemotaxis, Math. Nachr. 195 (1998) 77–114. [11] H. Gajewski, K. Zacharias, On a nonlocal phase separation model, J. Math. Anal. Appl. 286 (2003) 11–31. [12] G. Giacomin, J.L. Lebowitz, Phase segregation dynamics in particle systems with long range interactions. I. Macroscopic limits, J. Statist. Phys. 87 (1–2) (1997) 37–61. [13] G. Giacomin, J.L. Lebowitz, Phase segregation dynamics in particle systems with long range interactions. II. Interface motion, SIAM J. Appl. Math. 58 (6) (1998) 1707–1729. [14] O.A. Ladyzhenskaya, V.A. Solonnikov, N.N. Uraltseva, Linear and quasilinear equations of parabolic type, in: Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence RI, 1974. [15] W. Littman, G. Stampacchia, H.F. Weinberger, Regular points for elliptic equations with discontinuous coeMcients, Ann. Scuola Norm. Sup. Pisa 17 (3) (1963) 43–77. [16] F. Otto, L1 –contraction and uniqueness for quasilinear elliptic–parabolic equations, C.R. Acad. Sci. Paris 318 (1) (1995) 1005–1010. [17] I.V. Skrypnik, Methods for analysis of nonlinear elliptic boundary value problems, in: Translations of Mathematical Monographs, Vol 139, American Mathematical Society, Providence, RI, 1994, xii+348pp.
www.elsevier.com/locate/na
On unique solvability of nonlocal drift–di#usion-type problems H. Gajewskia;∗ , I.V. Skrypnikb a Weierstra-Institute
b Institute
for Applied Analysis and Stochastics, Mohrenstr. 39, Berlin D-10117, Germany for Applied Mathematics and Mechanics, Rosa Luxemburg Str. 74, 340114, Donetsk, Ukraine Received 14 May 2003; accepted 21 October 2003
Abstract We prove a priori estimates in L2 (0; T ; W 1; 2 ()) and L∞ (QT ), existence and uniqueness of solutions to Cauchy–Neumann problems for parabolic equations n @ @(u) @(u − v) (u) bi t; x; + a(t; x; v; u) = 0; (0.1) − @x @xi @t i=1 (t; x) ∈ QT = (0; T ) × ⊂ Rn+1 , where (u) = @(u)=@u ¿ 0 and the function v is de8ned by the nonlocal expression v(t; x) = − K(x; y)[(u(t; y)) − f(t; y)] dy; (0.2)
instead of solving an elliptic boundary problem as in the corresponding local case. Such problems arise as mathematical models of various di#usion–drift processes driven by gradients of local particle concentrations and nonlocal interaction potentials. An example is the transport of electrons in semiconductors, where u has to be interpreted as chemical and v as electro-statical potential. ? 2003 Elsevier Ltd. All rights reserved. MSC: 35B45; 35K15; 35K20; 35K65 Keywords: Nonlinear parabolic equations; Nonlocal drift; Bounded solutions; Uniqueness; Nonstandard assumptions; Degenerate type
∗
Corresponding author. E-mail address: [email protected] (H. Gajewski).
0362-546X/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2003.10.014
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1. Introduction We prove a priori estimates, existence and uniqueness of weak solutions to initial boundary value problems of the form n @(u − v) @ @(u) − + a(t; x; v; u) = 0 (t; x) ∈ QT ; (u)bi t; x; (1.1) @t @xi @x i=1 v(t; x) = − K(x; y)[(u(t; y)) − f(t; y)] dy (t; x) ∈ QT ; (1.2) n i=1
bi t; x;
@(u − v) @x
cos (; xi ) = 0
(t; x) ∈ = (0; T ) × @;
(1.3)
u(0; x) = h(x); x ∈ ; (1.4) u where (u) = 0 (s) ds, (u) ¿ 0, is a bounded open set in Rn and QT = (0; T ) × , T ¿ 0. In the case of smooth boundary @ of the set , is the outer unit normal on @ and (; xi ) is the angle between and the xi -axis. In the physical motivation and derivation (cf. [1,7]) of systems like (1.1)–(1.3) the ‘free energy’ c(t; y) −1 − f(t; y) dy d x; ( (c)) + c K(x; y) F(c) = 2 u (u) = s(s) ds (1.5) 0
plays an important role. Here c=(u) and ◦−1 can be seen as particle concentration and Legendre transformed of the primitive of , respectively, whereas the respective integrands model local and nonlocal particle interaction. Then, provided the reaction term a vanishes, system (1.1)–(1.4) describes the mass conserving evolution of c from the initial value c0 =(h) towards critical points or even minimizers of F under di#usion and drift forces, caused by the local and the global term in F, respectively. Moreover, the functional F will be also the key for our mathematical analysis of the system (cf. Theorem 1). Indeed, in the case that: the kernel K is symmetrical, the vector 8eld {bi (t; x; ·)} ∈ (Rn → Rn ) is monotone and bi (t; x; 0) = a = 0, we 8nd for solutions u; v of (1.1)–(1.3) according to the second law of thermodynamics @(u) dF(c) = (u − v) d x dt @t n @(u − v) @(u − v) bi t; x; d x 6 0; = − (u) @x @xi i=1
that means, F is Lyapunov functional in that case. Thus problems of the form (1.1)–(1.4) arise as intrinsically nonlocal mathematical models of various applied problems, for instance reaction–drift–di#usion processes of
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electrically charged species, phase transition processes and transport processes in porous media. The investigation of nonlinear nonlocal problems has received much attention in last years. In particular, in the papers [7,8,12,13] nonlocal models of phase separation were formulated and studied. One way to pass from (1.1)–(1.4) to local models consists in replacing the integral equation (1.2) by an elliptic di#erential equation like n @v @ (x) + (u) = f(t; x) (t; x) ∈ QT ; (1.6) − @xi @xi i=1
completed by some boundary condition for the function v. Such local problems were studied by many authors (cf. [6,10]). See also the papers [2,3,16], where degenerate parabolic equations were studied. Most strong results for local drift–di#usion type problems have been recently proved in [7]. Another way, applied in phase separation processes (cf. [4]), consists in replacing (1.5) by a local Ginsburg–Landau free energy expression 2
@c −1 F(c) = ( (c)) + c(1 − c) + d x; ; positive constants 2 @x and results in a fourth-order Cahn–Hilliard equation instead of the nonlocal second-order system (1.1)–(1.2). We consider problem (1.1)–(1.4) under standard conditions for the functions bi and some conditions for the function a to be formulated in Section 2. Our main speci8c assumption concerning Eq. (1.1) reads: (1 ) ∈ (R1 → R1 ) with (u) ¿ 0, u ∈ R1 , is continuous and has a piecewise continuous derivative such that (u)=(u) is nonincreasing on R1 . This condition seems natural in view of properties of probability particle distribution functions arising in mathematical physics. So in the semiconductor theory [1,6] relevant examples for functions satisfying condition (1 ) are given by = F!+1 , = = F! , where F! denotes the Fermi integral ∞ s! ds 1 F! (u) = ; ! ¿ − 1: (1.7) (! + 1) 0 1 + exp (s − u) Another example comes from phase separation problems [5,11], where the Fermi function 1 1 ; (1.8) ; (u) = (u) = (u) = −u u 1+e (1 + e )(1 + e−u ) plays a role corresponding to that of F!+1 . Our main assumption on the kernel K(x; y) is (K1 ) The function K(x; y) is de8ned for x; y ∈ and symmetrical, i.e., K(x; y) = K(y; x). Moreover, K(·; y) ∈ W 1; 1 () for almost every y ∈ such that @K(x; y) dy + ess sup @K(x; y) d x 6 –: (1.9) |K(x; y)| + ess sup @x @x x∈ y∈
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We remark that condition (K1 ) implies (cf. Lemma 1 below) properties as assumed in [5,11] for integral operators generated by kernels K(x; y) = K(|x − y|). Let us also remark that kernels |x − y|2−n ; log 1=|x − y|, corresponding to Newton potentials and fundamental solutions of Eq. (1.6) with bounded measurable function , satisfy condition (K1 ) [15]. The Green function for Eq. (1.6) satis8es condition (K1 ) in the cases of Dirichlet or Neumann boundary conditions for suMcient smooth @ and . Conditions on guarantying condition (K1 ) for the Green function can be formulated also in terms of smallness of the number −1 ess sup (x) ess inf (x) − 1: x∈
x∈
We formulate our assumptions and main results in Section 2. First, a priori estimates for the solution (u; v) are given in Section 3. In that section we prove also regularity properties of the function v, important for further considerations. An estimate of u in L∞ (QT ) is given in Section 4. Sections 5 and 6 are devoted to proofs of existence and uniqueness of solutions to problem (1.1)–(1.4) respectively. We are planning in forthcoming papers to apply our approach to systems of equations describing reaction–drift–di#usion processes in isothermal and non-isothermal cases. 2. Formulation of assumptions and main results Let be a bounded open set in Rn and QT = (0; T ) × , T ¿ 0. We assume that n ¿ 2. For n 6 2 it is necessary to make simple changes in our conditions that are connected with Sobolev’s embedding theorem. We assume the following condition on the set : (@) is such that the embeddings W 1; 1 () ⊂ Ln=(n−1) (), W 1; p () ⊂ L∞ () hold for p ¿ n. In view of the proof of a priori estimates for solutions of problem (1.1)–(1.4) we need restrictions on growth and on degeneration of the function as u → ±∞. From condition (1 ) the existence of $± = lim (u)
(2.1)
u→±∞
follows. For nonconstant functions (u) at least one of the limits $− , $+ is zero [8]. Studying the problem (1.1)–(1.4) we have to distinguish the cases of zero or non-zero values of $± . Therefore we shall consider two cases: ($1 ) $− = 0; $+ = 0;
($2 ) $− = $+ = 0:
Note that examples for ($1 ) and ($2 ) are given by (1.7) and (1.8), respectively. Our additional restrictions on the function are following: (2 ) if condition ($1 ) holds then positive constants 1 ; ! exist such that 2 ! ! −1 u ¿ 0; 0 6 ! ¡ ; 1 (u + 1) 6 (u) 6 1 (u + 1); n−1
(2.2)
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(3 ) there exists a positive constant 2 such that | (u)| 6 2 (u)
(2.3)
for u ¡ 0 in the case of condition ($1 ) and ∀u ∈ R1 if condition ($2 ) holds. Remark 1. The Fermi integrals F! (1.7) satisfy condition (2 ) for −1¡!¡(3 − n)= (n−1) and in particular for the physically most relevant case !=−1=2. In semiconductor modelling, especially in the classical drift–di#usion model, Fermi statistics is often approximated by Boltzmann statistics, i.e., F! is replaced by the exponential function. But, that approach violates condition 2 and leads to problems in proving uniqueness results at least in the three-dimensional case n = 3 and nonsmooth data. Let the functions a; bi from (1.1) satisfy the assumptions: (i) a(t; x; v; u), bi (t; x; &), i = 1; : : : ; n, are measurable with respect to t; x for every u; v ∈ R1 , & ∈ Rn and continuous with respect to u; v ∈ R1 , & ∈ Rn , for almost every (t; x) ∈ QT ; bi (t; x; 0) = 0, i = 1; : : : ; n; (ii) there exist positive constants 1 ; 2 such that ∀& ; & ∈ Rn and (t; x) ∈ QT n
[bi (t; x; & ) − bi (t; x; & )](&i − &i ) ¿ 1 |& − & |2 ;
i=1
|bi (t; x; &)| 6 2 (|&| + 1);
i = 1; : : : ; n;
(iii) there exist nonnegative functions $1 ∈ L1 (QT ), $ ∈ Lp1 (QT ), p1 ¿ (n + 2)=2, such that for arbitrary (t; x) ∈ QT , v; u ∈ R1 a(t; x; v; u)u ¿ 1 '(u)|u|m − 2 |v|m − $1 (t; x); |a(t; x; v; u)| 6 2 ('(u)|u| + |v|)m−1 + $(t; x); where m = (2 + !)=(1 + !) under condition ($1 ) and m = 2 under condition ($2 ), here '(u) is a nonnegative function bounded on R1 . We assume also an additional condition on the kernel K(x; y): (K2 ) if condition ($1 ) is satis8ed, then K(x; y)g(x)g(y) d x dy ¿ 0; ∀g ∈ L∞ ():
Remark 2. In relevant applications the kernel K models nonlocal particle interaction. Positive sign of K as assumed in condition (K2 ), corresponds to repulsive interaction between particles and implies, roughly speaking, global existence of solutions, whereas negative sign models attraction forces and may cause blow up of solutions (cf. [10]). However, under condition ($2 ) assumed in the papers [5,11] turns out to be bounded, so global existence can be proved without condition (K2 ).
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We consider problem (1.1)–(1.4) with f; h such that @f ∈ Lm (0; T; [W 1; m ()]∗ ); f ∈ C([0; T ]; Lp2 ()); @t h(x) ∈ L∞ ()
(2.4) (2.5)
and p2 ¿ n + 2!=(! + 1) in the case of condition ($1 ) and p2 ¿ n under condition ($2 ). Denition 1. A pair of functions (u; v), u; v ∈ L2 (0; T ; W 1; 2 ()), is called a solution of problem (1.1)–(1.4), if the following conditions are satis8ed: (i) the derivative @(u)=@t exists in the sense of distributions,
@u 2 @v 2 (u) + d x dt ¡ ∞; @x @x QT @(u) ∈ L2 (0; T ; [W 1; 2 ()]∗ ); @t (ii) ∀’ ∈ C ∞ (QO T ) and almost every , ∈ (0; T ), Q, = (0; ,) × , , n @(u − v) @’ @(u) (u)bi t; x; u; ; ’ dt + @t @x @xi 0 Q, i=1 (u) ∈ C([0; T ]; L2 ());
+ a(t; x; v; u)’
d x dt = 0;
equality (1.2) is satis8ed for almost all (t; x) ∈ QT ; (iii) ∀’ ∈ C ∞ (QO T ), satisfying ’(T; x) = 0 for x ∈ , T @’ @(u) ; ’ dt + d x dt = 0: [(u) − (h)] @t @t 0 QT
(2.6) (2.7)
(2.8)
(2.9)
Remark 3. Let (u; v) be a solution of problem (1.1)–(1.4). Since the space C ∞ (QO T ) is dense in the weighted space L2 (0; T ; W 1; 2 (; (u)), the integral identity (2.8) holds for all ’ ∈ L2 (0; T ; W 1; 2 ()) such that 2 @’ (u) d x dt ¡ ∞: @x QT
Remark 4. Lemma 1 below guarantees that the right-hand side of equality (1.2) is well-de8ned under our conditions on the functions and f. In what follows, we shall understand as known parameters all numbers from conditions (ii), (iii), (K1 ), norms of functions f; h; $1 , $ in respective spaces, numbers that depend only on ; T; n, the numbers 1 ; 2 ; 3 = max{(u): |u| 6 m0 } and 4 = min{(u) : |u| 6 m0 }, where m0 = h(x)L∞ () + 1: Further we shall denote by cj constants depending only on known parameters.
(2.10)
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Theorem 1. Let conditions (i) – (iii), (K1 ), (K2 ), (1 ), (@), (2.4), (2.5) be satis
@(u − v) 2 (u) dt d x 6 M1 ; ess sup (u(t; x)) d x + (2.11) @x t∈(0;T ) QT where
(u) =
0
u
s(s) ds:
(2.12)
Theorem 2. Let the assumptions of Theorem 1 and condition (2 ) be satis
@u 2 @v 2 (u) + d x dt 6 M2 : (2.13) @x @x QT Theorem 3. Let the assumptions of Theorem 2 be satis
(2.15)
Theorem 5. Let the conditions of Theorem 4 be satis
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Proposition 1. Let the conditions of Theorem 6 be satis
@u ∈ L∞ (0; T ; L2 ()) ∩ L2 (0; T ; W 1; 2 ()): @t
Remark 5. Proposition 1 and Theorem 4 imply that t → t @(u)=@t ∈ L∞ (0; T ; L2 ()). Consequently, (1.1) can be understood not only in the sense of distributions, but even as an equation in L2 (0; T ; L2 ()). Proofs of Theorems 1–3 are given in Section 3, proofs of Theorems 4–6 are given in Sections 4, 5, 6, respectively. 3. Integral estimates of the solution We start from auxiliary lemmas needed in the proofs of the Theorems 1–6. Let us de8ne operators K0 , K1 for g ∈ L∞ () by @K(x; y) |K(x; y)|g(y) dy; K1 g(x) = K0 g(x) = @x g(y) dy:
(3.1)
Lemma 1. The operators K0 , K1 are well de
for 1 6 p ¡ n;
for 1 6 p 6 ∞:
(3.2) (3.3)
Proof. Firstly, we prove (3.3). For p = 1 and ∞ (3.3) is a simple consequence of (1.9). For 1 ¡ p ¡ ∞ we 8nd for g ∈ Lp () by HPolder’s inequality p @K(x; y) p |K1 g(x)| d x 6 @x |g(y)| dy d x p−1 @K(x; y) @K(x; z) |g(y)|p dy 6 dx @x @x d z 6 –p |g(y)|p dy;
that is (3.3). For proving (3.2) we use the embedding theorem for W 1; 1 () to infer from (1.9) (3.4) ess sup |K(x; y)|n=(n−1) dy 6 c1 : x∈
H. Gajewski, I.V. Skrypnik / Nonlinear Analysis 56 (2004) 803 – 830
811
Now by HPolder’s inequality we have for 1 6 p ¡ n, g ∈ L∞ (), |K0 g(x)|np=(n−p) d x
6
|K(x; y)|n(p−1)=((n−1)p) · [|K(x; y)|n−p=((n−1)p)
(n−p)=n
× |g(y)| 6
×
] · |g(y)|
|K(x; y)|n=(n−1) dy
p=n
|g(y)| ˜ p d y˜
np=(n−p) dy
dx
(n(p−1))=(n−p) · |K(x; y)| O n=(n−1) · |g(y)| O p d yO
p=(n−p) d x 6 c2
|g(x)|p d x
n=(n−p) :
This inequality implies (3.2) and the proof of Lemma 1 is complete. Lemma 2. Let the assumptions of Theorem 1 be satis
(3.5)
holds for each , ∈ (0; T ) with a constant M5 depending only on known parameters. Proof. Let , ∈ (0; T ) and de8ne for 0 ¡ 2 ¡ T − , , [(u(t + 2; x)) · v(t + 2; x) − (u(t; x))v(t; x)] d x dt: I (2) = 0
(3.6)
By writing I (2) as di#erence of two integrals and changing the integration variable in the 8rst integral we get ,+2 2 I (2) = (u(t; x))v(t; x) d x dt − (u(t; x))v(t; x) d x dt: (3.7) ,
0
On the other hand, we can rewrite I (2) as I (2) = I1 (2) + I2 (2); where I1 (2) = I2 (2) =
, 0
0
,
[(u(t + 2; x)) − (u(t; x))]v(t + 2; x) d x dt; (u(t; x))[v(t + 2; x) − v(t; x)] d x dt:
(3.8)
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H. Gajewski, I.V. Skrypnik / Nonlinear Analysis 56 (2004) 803 – 830
Using (1.2) and setting v1 (t; x) = K(x; y)(u(t; y)) dy, we can rewrite I2 (2) as , [(u(t + 2; x)) − (u(t; x))]v(t; x) d x dt I2 (2) = 0
+
, 0
2
,
−
−
,+2
,
[f(t + 2; x) − f(t; x)]v1 (t; x) d x dt [f(t − 2; x) − f(t; x)]v1 (t; x) d x dt
f(t − 2; x)v1 (t; x) d x dt +
0
2
f(t; x)v1 (t; x) d x dt:
(3.9)
From (3.8)–(3.9), (2.4), (2.7) and Lemma 1 we see that dividing I (2) by 2 and passing to the limit 2 → +0 gives (u(,; x))v(,; x) d x − (h(x))v(0; x) d x
=2
, 0
−
@(u) ;v @t
dt + 2
, 0
f(,; x)v1 (,; x) d x +
@f ; v1 @t
dt
f(0; x)v1 (0; x) d x:
(3.10)
We shall estimate the summands in (3.10). In the case of condition ($1 ) we have by (2.3), (3.2) and condition (K2 ) (u(,; x))v(,; x) d x
=
6
f(,; x)v1 (,; x) −
K(x; y)(u(,; x))(u(,; y)) dy
f(,; x)v1 (,; x) d x 6 c3 (u(,; x)L1 () :
dx (3.11)
An analogous estimate is true under condition ($2 ), because of the boundedness of the function in that case. Further, using Lemma 1 we get , , 1=m @v1 m @f m ; v1 dt 6 c4 @x + v1 d x dt @t 0 0 6 c5 (u)Lm (Q, ) :
(3.12)
Estimating the remaining summands in (3.10) and using (3.11), (3.12), we obtain from (3.10) the desired estimate (3.5) and the proof of Lemma 2 is complete. Proof of Theorem 1. Condition (2.6) and Remark 3 allow us to use the test function ’ = u − v in integral identity (2.8). Then, evaluating the resulting terms by conditions
H. Gajewski, I.V. Skrypnik / Nonlinear Analysis 56 (2004) 803 – 830
813
(ii), (iii) and Lemma 2, we obtain , @(u − v) 2 @(u) dx (u) ; u dt + @t @x 0 Q, + '(u)|u(t; x)|m d x dt 6 c6 1 + (u(,; x))L1 () Q,
+ (u)Lm (Q, ) +
Q,
[|v(t; x)|m + $1 (t; x) + $1+! (t; x)] d x dt :
We transform the 8rst integral in (3.13) in the following way: , @(u) (u; (,; x)) d x − (h(x)) d x ; u dt = @t 0
(3.13)
(3.14)
with (u) de8ned by (2.9). The proof of equality (3.14) is analogous to the proof of Lemma 1 in [9]. Remarking that condition (2 ) implies c7−1 [(u)]m − c8 6 (u) 6 c7 [(u)]m + c8 ;
(3.15)
using (3.14) and Lemma 1, we obtain from (3.13) @(u − v) 2 m d x dt (u) (u(,; x)) d x + @x Q, + '(u)|u(t; x)|m d x dt 6 c9 1 + m (u(t; x)) d x dt : Q,
Q,
(3.16)
Now estimate (2.11) follows from (3.15), (3.16) and Gronwall’s Lemma and the proof of Theorem 1 is complete. Proof of Theorem 2. The assertion of Theorem 2 follows simply under ($2 ). Indeed, in this case the functions ; are bounded such that (2.4) and Lemma 1 imply @v=@x ∈ C([0; T ], Lp2 ()). Hence (2.13) follows immediately from (2.11). Let us now assume that condition ($1 ) is satis8ed. In this case we de8ne u+ (t; x) = max{u(t; x); 0};
QT± = {(t; x) ∈ QT : ± u(t; x) ¿ 0}:
(3.17)
Theorem 1 implies [1 + u+ (t; x)]!+2 d x dt ess sup t∈(0;T )
+
QT+
@(u − v) 2 d x dt 6 c10 : [1 + u(t; x)] @x !
Thus for proving the desired inequality (2.13), it suMces to show that 2 @v [1 + u+ (t; x)]! d x dt 6 c11 : @x QT
(3.18)
(3.19)
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H. Gajewski, I.V. Skrypnik / Nonlinear Analysis 56 (2004) 803 – 830
De8ne for q ¿ 2=(! + 1), ¿ ! + 2, q ! @v [1 + u+ (t; x)] d x dt; I (q) = @x QT
J( ) =
QT
u+ (t; x) d x dt:
(3.20)
We shall need the following assertions: (1) the estimate I (2=(! + 1)) 6 c12 holds; O 6 c14 with qO = ( O − !)=(! + 1); (2) if for O ¿ ! + 2, J ( O) 6 c13 , then I (q) (3) if for q˜ ∈ [2=(! + 1); 2], I (q) ˜ 6 c15 , then J ( ˜) 6 c16 , with ˜ = q˜ + ! + (q=n)(2 + !). To prove assertion (1) we apply (1.9), Theorem 1, HPolder’s and Young’s inequalities: @K(x; y) 2 ! (2=(!+1))−1 6– [1 + u+ (t; x)] I @y !+1 QT ×|(u(t; y)) − f(t; y)|2=(!+1) dy d x dt 6 c17 –(2=(!+1)) {1 + [u+ (t; x)]2! + |f(t; x)|m } d x dt; QT
(3.21)
where we used also (2.2) and the simple inequality |(u)| 6 c18 (1 + u+ )!+1 ;
u+ = max(u; 0):
(3.22)
Now (3.21), (3.18) and (2.4) imply assertion (1). Assertion (2) follows from the next inequality that is obtained analogously to (3.21). @K(x; y) ! qO q−1 O [1 + u+ (t; x)] I (q) O 6– @y |(u(t; y)) − f(t; y)| dy d x dt QT O q {1 + [u+ (t; x)]!+q(!+1) + |f(t; x)|}!=(!+1)+qO d x dt 6 c20 : (3.23) 6 c19 – QT
Assertion (3) follows by HPolder’s inequality and Sobolev’s embedding theorem. Indeed, we get from (3.18) q=n T ˜ q˜ 2+! [1 + u+ (t; x)] d x J q˜ + ! + (2 + !) 6 n 0 (n−q)=n ˜ (q+!)n=(n− ˜ q) ˜ × [1 + u+ (t; x)] dx dt
6 c21
QT+
q˜ @u [1 + u(t; x)]! d x dt @x
+ c21
QT
2+!
[1 + u+ (t; x)]
(!+q)=(!+2) ˜ d x dt
:
(3.24)
H. Gajewski, I.V. Skrypnik / Nonlinear Analysis 56 (2004) 803 – 830
Inequalities I (q) ˜ 6 c15 and (3.18) imply q˜ ! @u [1 + u(t; x)] d x dt 6 c22 ; + @x QT
815
(3.25)
we obtain assertion (3) from (3.24), (3.25) and (3.18). Let us de8ne sequences {qi }, { i }, i = 1; : : : ; N , such that q1 =
2 ; !+1
i
qN −1 ¡ 2;
= qi + ! +
qi (2 + !); n
qi+1 =
i −! ; !+1
qN ¿ 2:
(3.26)
This de8nition is justi8ed by (2.2) and qi 2 [2 − !(n − 1)] ¿ 0: [2 − !(n − 1)] ¿ n(! + 1)2 n(! + 1)
qi+1 − qi =
Now, using the assertions (1) – (3), we get by iteration that I (qN ) 6 c23 and hence (3.19). This ends the proof of Theorem 2. Proof of Theorem 3. Analogously as in the proof of Theorem 2, we can restrict us to the case of condition ($1 ). We test integral identity (2.8) with ’(t; x) = [(uk (t; x)) − 0 ]+ {1 + [(uk (t; x)) − 0 ]2 }r ;
(3.27)
where uk (t; x) = min {u(x; t); k}, k ¿ m0 , m0 is given by (2.10), 0 = (m0 ) and r ∈ (− 1=2; ∞) is an arbitrary number. Analogously to Lemma 1 in [9] we have , @(u) (r) (u(,; x)) d x; (3.28) ; ’ dt = @t 0 where (r) (u) =
0
u
(s)[(sk ) − 0 ]+ {1 + [(sk ) − 0 ]2 }r ds;
sk = min {s; k} (3.29)
and (r) (u) ¿
1 {1 + [(uk ) − 0 ]2 }r+1 − 1 2(r + 1)
for u ¿ m0 :
(3.30)
We write the derivative of ’ in the form @’ @u = (u)8(r) (uk ) 9(m0 ¡ u ¡ k); @xi @xi
(3.31)
where 9(m0 ¡ u ¡ k) is the characteristic function of the set {(t; x) ∈ QT : m0 ¡ u(t; x) ¡ k} and the function 8(r) (u) satis8es c23 r{1 + [(u) − 0 ]2 }r 6 8(r) (u) 6 c24 (r + 1){1 + [(u) − 0 ]2 }r
(3.32)
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H. Gajewski, I.V. Skrypnik / Nonlinear Analysis 56 (2004) 803 – 830
with r = min (1 + 2r; 1). Using (3.28)–(3.32) and conditions (ii), (iii), we obtain {1 + [(uk (,; x)) − 0 ]2 }r+1 9(m0 ¡ u) d x
+
2 @u 2 (u){1 + [(uk ) − 0 ]2 }r · 9(m0 ¡ u ¡ k) d x dt @x 0
2 2 , @v r+1 6 c24 2 (u){1 + [(uk ) − 0 ]2 }r r @x 0
,
×9(m0 ¡ u ¡ k) d x dt +
r+1 [|u|m−1 + |v|m−1 + $(t; x)] 1+ r Q,
× {1 + [(uk ) − 0 ]2+ }r+1=2 9(m0 ¡ u) d x dt
:
(3.33)
We introduce the notations {u ¿ 1} = {(t; x) ∈ QT : u(t; x) ¿ 1} and 2 @u I ∗ (q) = ess sup q (u+ (t; x)) d x + 2 (u)q−2 (u) d x dt; @x t∈(0;T ) {u¿1} 2+! [1 + u+ (t; x)] d x dt; q ¿ J ∗( ) = ; ¿ ! + 2: (3.34) 1 +! QT We shall need the following assertions: (1) I ∗ ((2 + !)=(1 + !)) 6 c25 ; (2) if J ∗ ( O) 6 c26 , O ¿ ! + 2, then I ∗ (q) O 6 c27 , qO = min {( O − 2!)=(! + 1); p2 − 2!= (! + 1); O=(1 (! + 1)) + 1}; (3) if I ∗ (q) ˜ 6 c28 for q˜ ¿ (2+!)=(1+!), then J ∗ ( ˜) 6 c29 for ˜ =(1=n)q(n+2)(1+!). ˜ Remarking that 2 (u)q−2 (u) 6 c30 (u) for u ¿ 1, q = (2 + !)=(1 + !), we obtain assertion (1) immediately from Theorems 1 and 2. To prove the assertion (2) we start estimating the 8rst integral on the right-hand side of (3.33) with r 6 min {( O − 2!)=(2(! + 1)) − 1, (p2 =2) − (!=(! + 1)) − 1}. Analogously to inequality (3.21) we have 2 , 2 r @v 2 (u){1 + [(uk ) − 0 ] } 9(m0 ¡ u ¡ k) d x dt @x 0 2 2!+2r(1+!) @v [1 + uk (t; x)] 6 c31 @x ) d x dt Q,+ @K(x; y) 2!+2r(1+!) 6 c31 – [1 + uk (t; x)] @y Q,+ ×|(u(t; y)) − f(t; y)|2 dy d x dt
H. Gajewski, I.V. Skrypnik / Nonlinear Analysis 56 (2004) 803 – 830
6 c32
QT
6 c33
817
{1 + |(u(t; x))| + |f(t; x)|}2!=(!+1)+2r+2 d x dt
{1 + [u+ (t; x)]!+1 + |f(t; x)|}2!=(!+1)+2r+2 d x dt 6 c34 :
QT
(3.35)
Let us now estimate the last integral in (3.33). Using Theorem 1, Lemma 1, HPolder’s inequality and supposing r such that m m 6 O; (2r + 1)(! + 1)p1 6 O; 6 p2 ; 2(! + 1) r + 2 r+ 2 2 we obtain Q,+
[|u|m−1 + |v|m−1 + $(t; x)]{1 + [(uk ) − 0 ]2+ }r+(1=2) d x dt
6 c34
Q,+
+
+
(|(u)| + |f(t; x)|)2(r+(m=2)) d x dt
Q,
[1 + uk ]2(!+1)(r+(m=2)) dt d x
[1 + uk ](2r+1)(!+1)p1 d x dt
Q,+
1=p1
6 c35 :
(3.36)
From (3.35) and (3.36) we see that the left-hand side in (3.33) is bounded by some constant depending only on known parameters and independent of k; r, provided J ∗ ( O) 6 c26 and r is de8ned by
O − 2! O 2! 1 r = min + 1 − 1: (3.37) ; p2 − ; 2 !+1 ! + 1 p1 (! + 1) O 6 c27 . That is So we are able to pass to the limit k → +∞ in (3.33) to obtain I ∗ (q) assertion (2). Assertion (3) follows from HPolder’s inequality and Sobolev’s embedding theorem analogously to inequality (3.24). Now we de8ne numbers {qi }, { i }, i = 1; : : : ; N , such that q1 =
2+! ; 1+!
i
=
1 qi (n + 2)(1 + !); n
qi+1 = min
− 2! 2! i +1 ; ; p2 − ; 1+! 1 + ! p1 (! + 1)
qN −1 ¡ p2 −
2! ; !+1
i
qN = p2 −
2! : !+1
(3.38)
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H. Gajewski, I.V. Skrypnik / Nonlinear Analysis 56 (2004) 803 – 830
This de8nition is justi8ed, since {qi } is increasing by − 2! 2qi 2! 2 − qi = − ¿ [! + 2 − n!] ¿ 0; !+1 n !+1 n(! + 1) p1 − 1 n + 2 i = q · + 1 − q − 1 + 1 ¿ 1: i i p1 (! + 1) p1 n i
(3.39)
Note also that N ¿ (n + 2)(1 + !). So assertions (1) – (3) imply I ∗ (qi ) 6 c36 , J ∗ ( i ) 6 c37 for i = 1; : : : ; N . By I ∗ (qN ) 6 c36 , J ∗ ( N ) 6 c37 we have [(u(t; x))]p2 −2!=(!+1) d x 6 c36 ; ess sup t∈(0;T )
QT
|(u(t; x))|
N =(1+!)
d x dt 6 c37 :
(3.40)
Hence conditions (2.4), (’) and Lemma 1 imply (2.14) with p3 = min {p2 − 2!= (! + 1); N =(! + 1) − 2} and the proof of Theorem 3 is complete.
4. Boundedness of the function u Firstly we want to estimate u(t; x) from above under condition ($1 ). Lemma 3. Let the conditions of Theorem 4 and ($1 ) be satis
(4.1)
Proof. We apply (3.33) and estimate the integrals on the right-hand side by HPolder’s inequality. Using the properties of the function $, (2.14) and (3.40), we get 2 @v 2 (u){1 + [(uk ) − 0 ]2 } 9(m0 ¡ u ¡ k) d x dt @x Q, + [|u|m−1 + |v|m−1 + $(t; x)]{1 + [(uk ) − 0 ]2 }r+1=2 9(m0 ¡ u) d x dt Q,
6 c39
Q,
2 (r+1)pO
{1 + [(uk ) − 0 ] }
1=pO 9(m0 ¡ u) d x dt
(4.2)
with pO ¡ (n + 2)=n depending only on known parameters. Inequalities (3.33), (4.2) imply for r ¿ 1 {1 + [(uk (,; x)) − 0 ]2 }r+1 9(m0 ¡ u) d x
H. Gajewski, I.V. Skrypnik / Nonlinear Analysis 56 (2004) 803 – 830
2 @u 2 (u){1 + [(u) − 0 ]2 }r 9(m0 ¡ u ¡ k) d x dt @x Q, 1=pO 2 (r+1)pO 2 6 c40 r 1 + {1 + [(uk ) − 0 ] } 9(m0 ¡ u) d x dt :
819
+
Q,
(4.3)
Last estimate, HPolder’s inequality and Sobolev’s embedding inequalities yield for r ¿ 1 {1 + [(uk ) − 0 ]2 }(r+1)(n+2)=n · 9(m0 ¡ u) d x dt QT
6 c41 · ×
T
0
2 r+1
{1 + [(uk ) − 0 ] }
2=n · 9(m0 ¡ u) d x
{1 + [(uk ) − 0 ]2 }(r+1)n=(n−2) d x
6 c42 r 2 ess sup
0¡t¡T
(n−2)=n dt
{1 + [(uk (t; x)) − 0 ]2 }r+1 9(m0 ¡ u) d x
2=n
2 @u 2 (u){1 + [(u) − 0 ]2 }r 9(m0 ¡ u ¡ k) @x QT +{1 + [(uk ) − 0 ]2+ }r+1 d x dt
×
6 c43 r
4+4=n
QT
{1 + [(uk ) −
0 ]2+ }(r+1)pO
(1+(2=n))1=pO d x dt
:
(4.4)
Inequalities (4.4) and (3.40) justify the application of Moser’s iteration process to verify (4.1) and the proof of Lemma 3 is complete. For arbitrary k ∈ R and functions w on QT we de8ne w(k) = w(k) (t; x) = max {w(t; x); k};
w− = w− (t; x) = min{w(t; x); 0}:
(4.5)
Lemma 4. Let the conditions of Theorem 4 be satis
(4.7)
820
H. Gajewski, I.V. Skrypnik / Nonlinear Analysis 56 (2004) 803 – 830
Proof. We test the integral identity (2.8) with 1 ’= [(u(k) ) − (−m0 )]− · |u(k) + m0 |r ; k ¡ − m0 ; r ¿ 0: (u(k) ) Then, analogously to the proof of inequality (4.32) in [7], we obtain |[u(k) (,; x) + m0 ]− |r+1 d x
+
Q,
2 @u |u + m0 | 9(k ¡ u ¡ − m0 ) d x dt @x r
2 2 2
@u @v r @v 6 c44 (r + 1) |u + m0 | + + @x @x @x Q, 2
× 9(k ¡ u ¡ − m0 ) + [1 + $(t; x)] · |[u(k) + m0 ]− |r
d x dt:
(4.8)
Using Lemma 4 we have from (4.8) 2 @u |u + m0 |r 9(k ¡ u ¡ − m0 ) d x dt |[u(k) (,; x) + m0 ]− |r+1 d x + @x Q, 2 (k) r 6 c45 (r + 1) |[u + m0 ]− | $(t; ˜ x) d x dt + 1 ; (4.9) Q,
where $(t; ˜ x)=$(t; x)+|@v=@x|2 +1. The condition on $ and Theorem 3 imply $˜ ∈ Lp˜ (QT ) with p˜ ¿ (n + 2)=2. Inequality (4.9) allows to apply Moser’s iteration process for proving |[u(k) (t; x) + m0 ]− | 6 c46 : This implies (4.7) with M8 = m0 + c46 and Lemma 5 is proved. Proof Theorem 4. The assertion of Theorem 4 follows immediately from the Lemmas 3, 5 if condition ($1 ) is satis8ed. In the case of condition ($2 ) Lemma 5 yields a lower bound of u(t; x). The existence of an upper bound in that case can be analogously shown. The proof of Theorem 4 is complete. 5. Proof of the existence theorem Firstly we shall assume that condition ($1 ) is satis8ed. In this case we regularize the problem (1.1)–(1.4) by replacing ; a; by ∗ ; a∗ ; ∗ in the following way: Let M4 be the constant from Theorem 4 and (t; x) ∈ QT , v ∈ R1 , then ∗ (u) = (u); a∗ (t; x; v; u) = a(t; x; v; u); ∗ (u) = (u)
if u 6 M4 ;
(5.1)
∗ (u) = (M4 )eM4 −u ; a∗ (t; x; v; u) = a(t; x; v; M4 )eM4 −u ; ∗ (u) = (M4 ) + (M4 )[1 − eM4 −u ]
if u ¿ M4 :
(5.2)
H. Gajewski, I.V. Skrypnik / Nonlinear Analysis 56 (2004) 803 – 830
We consider the regularized problem in QT , i.e., n @(u − v) @ @∗ (u) + a∗ (t; x; v; u); ∗ (u)bi t; x; − @x @t @x i c=i v(t; x) = − K(x; y)[∗ (u(t; y)) − f(t; y)] dy; n
bi
i=i
@(u − v) t; x; @x
u(0; x) = h(x);
821
(5.3) (5.4)
cos (; xi ) = 0
(t; x) ∈ (0; T ) × @;
x ∈ :
(5.5) (5.6)
This problem satis8es all conditions of Section 2 with the same known parameters as problem (1.1)–(1.4). Therefore each solution (u; v) of problem (5.3)–(5.6) satis8es a priori estimate (2.15). So from (5.1) we see that a solution (u; v) of problem (5.3)–(5.6) is automatically solution of problem (1.1)–(1.4). Therefore it is suMcient to establish the existence of a solution of problem (5.3)–(5.6) in order to prove the Theorem 5. Let X (k); k ∈ [0; 1], be the Banach space of functions such that |u(t + 2; x) − u(t; x)|2 d x dt ¡ ∞: u2X (k) = u2L2 (0; T ;W 1; 2 ()) + sup 2k 0¡2¡T=2 QT −2 To study the solvability of problem (5.3)–(5.6) we introduce the operator A : X (1=2) → X (1=2) transforming a function g ∈ X (1=2) into the solution U = Ag of the following problem in QT : n @(U − G) @∗ (U ) @ + a∗ (t; x; G; U ) = 0; ∗ (U )bi t; x; (5.7) − @t @xi @x i=1 G(t; x) = − K(x; y)[∗ (g(t; y)) − f(t; y)] dy; (5.8) n i=1
bi
@(U − G) t; x; @x
U (0; x) = h(x);
cos(; xi ) = 0;
x ∈ :
(t; x) ∈ (0; T ) × @;
(5.9) (5.10)
Taking into account the boundedness of the function ∗ , assumptions (2.4), (@) and Lemma 1 we have @G(t; x) p2 (5.11) ess sup @x d x + ess sup |G(t; x)| 6 c47 ; t∈(0;T )
(t;x)∈QT
with a constant c47 depending only on known parameters and independent of g. In order to guaranty the unique solvability of problem (5.7), (5.9), and (5.10) for given function G satisfying (5.11), the Theorems 3, 4 in [9] can be adapted. Indeed, the functions @G(t; x) ; i = 1; : : : ; n b∗i (t; x; &) = bi t; x; & − @x
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H. Gajewski, I.V. Skrypnik / Nonlinear Analysis 56 (2004) 803 – 830
satisfy the inequalities n [b∗i (t; x; & ) − b∗i (t; x; & )](&i − &i ) ¿ |& − & |2 ;
(5.12)
i=1
|b∗i (t; x; &)| 6 2 |&| + ?(t; x)
(5.13)
with ?(t; x) = 2 (1 + |@G=@x|) ∈ L∞ (0; T ; Lp2 ()), which essentially coincide with conditions (ii) ensuring in [9] existence and uniqueness in the case of Dirichlet boundary conditions. But it is simple to check that the Theorems 3 and 4 in [9] are also true for Neumann boundary conditions. Estimate (5.11) and adaptations of the Theorems 1 and 2 from [9] imply @U (t; x) 2 ess sup {|U (t; x)|: (t; x) ∈ QT } 6 M9 ; (5.14) @x d x dt 6 M9 ; QT where U (t; x) is the solution of problem (5.7), (5.9), (5.10) and M9 is a constant depending only on known parameters and independent of g. Using estimates (5.14) and (5.11) we can show analogously to [14] that |U (t + 2; x) − U (t; x)|2 sup d x dt 6 M10 ; (5.15) 2 0¡2¡T=2 QT −2 with a constant M10 depending only on known parameters and independent of g. So the solution U of problem (5.7), (5.9), (5.10) belongs to the space X (1) and therefore the operator A : X (1=2) → X (1=2) is well de8ned. From the de8nition of this operator we see immediately that the solvability of problem (5.3)–(5.6) is equivalent to the existence of a 8xed point of the operator A Ag = g; g ∈ X 12 : (5.16) We shall prove the existence of a solution of (5.16) by using the Leray–Schauder principle. The Leray–Schauder degree theory implies (cf. [14,17]) that for the solvability of Eq. (5.16) it is suMcient to establish following statements: (1) there exists a family {A@ }, @ ∈ [0; 1], of operators A@ : X (1=2) → X (1=2) such that A1 = A, A0 = 0, A@ is completely continuous for each @ ∈ [0; 1] and {A@ } satis8es following continuity condition: for arbitrary sequences {@j }, {uj } such that @j → @0 , uj → u0 we have A@j uj → A@0 u0 , where → denotes strong convergence in X (1=2). (2) there exists a positive number R such that A@ g = g
for @ ∈ [0; 1];
g
1 X(2)
= R:
(5.17)
We de8ne A@ g = U@ , where U@ is the solution of the problem n @∗ (U@ ) @ @(U@ − @G) ∗ − (U@ )bi t; x; @t @xi @x i=1
+ a∗ (t; x; G; U@ ) − (1 − @)a∗ (t; x; G; 0) = 0; (t; x) ∈ QT ;
(5.18)
H. Gajewski, I.V. Skrypnik / Nonlinear Analysis 56 (2004) 803 – 830 n
bi
i=1
@(U@ − @G) t; x; @x
U@ (0; x) = @h(x);
823
cos (; xi ) = 0;
(t; x) ∈ (0; T ) × @;
x ∈ :
(5.19) (5.20)
The function G(t; x) in (5.18) – (5.20) is given by (5.8). The unique solvability of this problem can be seen as that of (5.7), (5.9), (5.10). Hence the operator A@ : X (1=2) → X (1=2) is well de8ned. We shall check 8rstly statement (2) formulated above. Let us assume that @; g are such that @ ∈ [0; 1], g ∈ X (1=2) and A@ g = g. Then from (5.18)–(5.20), (5.8) we see that the pair (U; G) is a solution of a nonlocal nonlinear problem being analogous to problem (1.1)–(1.4). Consequently, by Theorem 2 there exists a constant M11 depending only on known parameters and independent of @ ∈ [0; 1] such that U@ L2 (0; T ;W 1; 2 ()) 6 M11 . From the corresponding inequality (5.15) we have U@ X (1=2) 6 M12 with a constant M12 depending only on known parameters and independent of g(t; x); @. Since the equality A@ g=g implies gX (1=2) 6 M12 , the desired relation (5.17) is ful8lled for R = M12 + 1. Now we shall check statement (1) formulated above. The equalities A1 = A and A0 = 0 hold because of the unique solvability of problem (5.7), (5.9), (5.10). Thus it remains to prove compactness and continuity of the operator A. To this aim we prove the following lemma. Lemma 6. Assume that the conditions of Theorem 4 are satis
We put the test functions ’i given by ’1 =
1 [∗ (U1 ) − ∗ (U2 )]; ∗ (U1 )
’2 = U1 − U2 ;
into the integral identities corresponding to Ui . Taking the di#erence of the two resulting equalities, we get ∗ , ∗ @ (U1 ) @ (U2 ) 1 dt ; ∗ [∗ (U1 ) − ∗ (U2 )] − ; U1 − U2 @t (U1 ) @t 0 n @(U1 − G1 ) @ 1 ∗ ∗ + ( ∗ (U1 )bi t; x; (U ) − (U )) 1 2 @x @xi ∗ (U1 ) Q, i=1
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H. Gajewski, I.V. Skrypnik / Nonlinear Analysis 56 (2004) 803 – 830
∗
− (U2 )bi
@(U2 − G2 ) t; x; @x
@ [U1 − U2 ] @xi
d x dt
∗ (U1 ) − ∗ (U2 ) ∗ (U1 ) Q, − a∗ (t; x; G2 ; U2 )(U1 − U2 ) d x dt = 0: a∗ (t; x; G1 ; U1 )
+
(5.22)
We transform the 8rst integral in (5.22) analogously to Lemma 2 in [9] to obtain ∗ , ∗ @ (U1 ) @ (U2 ) 1 ∗ ∗ dt ; ∗ [ (U1 ) − (U2 )] − ; U1 − U2 @t (U1 ) @t 0 U1 (,;x) ∗ = [U1 (,; x) − s] (s) ds d x ¿ c48 |U1 (,; x) − U2 (,; x)|2 d x: (5.23)
U2 (,;x)
To estimate the second integral in (5.22), we note that by condition (1 ) U1 ∗ ( ) (s) ∗ (∗ ) (U1 ) ∗ ∗ (s) ds − [ (U1 ) − (U2 )] ¿ − ∗ (s) ∗ (U1 ) U2 = ∗ (U2 ) − ∗ (U1 ); such that n
∗
(U1 )bi
i=1
¿
@(U1 − G1 ) t; x; @x
@ @xi
1 (∗ (U1 ) − ∗ (U2 )) ∗ (U1 )
(5.24)
n @U1 ∗ @(U1 − G1 ) @G1 − bi t; x; − bi t; x; [ (U2 ) − ∗ (U1 )] @x @x @xi i=1
n @(U1 − G1 ) @U1 @U2 ∗ ∗ (U1 ) bi t; x; − (U2 ) + @x @xi @xi i=1
n @G1 @U1 (∗ ) (U1 ) ∗ − bi t; x; − · ∗ [ (U1 ) − ∗ (U2 )]: @x @xi (U1 )
(5.25)
i=1
Since the properties of the function ensure that ∗ ∗ (U1 ) − ∗ (U2 ) − ( ) (U1 ) · [∗ (U1 ) − ∗ (U2 )] 6 c49 |U1 − U2 |; ∗ (U1 ) we get from (5.11), (5.25), (5.26) and condition (ii) n @(U1 − G1 ) @ 1 ∗ ∗ ∗ (U1 )bi t; x; ( (U1 ) − (U2 )) @x @xi ∗ (U1 ) i=1
∗
− (U2 )bi
@(U2 − G2 ) t; x; @x
@ (U1 − U2 ) @xi
(5.26)
H. Gajewski, I.V. Skrypnik / Nonlinear Analysis 56 (2004) 803 – 830
825
2 @(U1 − U2 ) 2 − c51 @(G1 − G2 ) + 1 + @G1 @U1 ¿ c50 @x @x @x @x × |U1 − U2 | :
(5.27)
Using condition (iii) and (5.14), we can estimate the last integral in (5.22) ∗ ∗ ∗ a (t; x; G1 ; U1 ) (U1 ) − (U2 ) − a∗ (t; x; G2 ; U2 )(U1 − U2 ) ∗ (U1 ) 6 c52 |U1 − U2 | [1 + $(t; x)]:
(5.28)
Finally, from (5.22), (5.23), (5.27) and (5.28) we see that @(U1 − U2 ) 2 2 d x dt |U1 (,; x) − U2 (,; x)| d x + @x Q,
@(G1 − G2 ) 2 @G1 @U1 6 c53 + 1 + @x @x · |U1 − U2 | @x Q, +[1 + $(t; x)] |U1 − U2 |
d x dt:
(5.29)
Now we are ready to return to the study of properties of the operator A. We begin with the compactness. Let {gj } be a bounded sequence in X (1=2). Then by the compactness of the embedding X (1=2) ⊂ L2 (QT ) we can assume that {gj } converges strongly in L2 (QT ) to some function g0 . This and Lemma 1 imply the strong convergence of @Gj =@x to @=@xG0 in [L2 (QT )]n , where Gj is de8ned analogously to (5.21). Using (5.14), (5.15) with Uj = Agj , we can assume that Uj converges to some U0 ∈ X (1=2) weakly in L2 (0; T ; W 1; 2 ()) and strongly in Lq (QT ) for an arbitrary q ¡ ∞. In order to prove strong convergence of {Uj } in L2 (0; T ; W 1; 2 ()), we use (5.29) with U1 = Uj , U2 = Ui , G1 = Gj , G2 = Gi and we obtain @(Uj − Ui ) 2 d x dt |Uj (,; x) − Ui (,; x)|2 d x + ess sup @x ,∈(0;T ) QT
@(Gj − Gi ) 2 @Gj @Uj 6 c54 + 1 + @x · @x · |Uj − Ui | @x QT + [1 + $(t; x)]|Ui − Uj |
d x dt:
(5.30)
Using already known convergence properties of the sequences {Uj }, {Gj } and (5.11), we see that the right-hand side of (5.30) tends to zero as j; i → ∞. That means compactness of the sequence {Uj } in L2 (0; T ; W 1; 2 ()). The compactness of this sequence in X (1=2) follows now from (5.30) and (5.15) with Uj , Ui . So we have established the compactness of the operator A. Now we shall check its continuity. Let {gj } be a sequence converging strongly in X (1=2) to g0 . Lemma 1 implies that
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(@=@x)Gj → (@=@x)G0 in [L2 (QT )]n . Using the compactness of A we can assume that {Uj = Agj } converges strongly in X (1=2) to some UO 0 ∈ X (1=2). We have to show UO 0 = Ag0 . From the integral identity for Uj , ∗ n @(Uj − Gj ) @’ @ (Uj ) ; ’ dt + ∗ (Uj )bi t; x; @t @x @xi 0 Q, i=1 + a∗ (t; x; Gj ; Uj )’
d x dt = 0;
’ ∈ L2 (0; T ; W 1; 2 ())
(5.31)
we obtain the boundedness of the sequence {(@=@t)∗ (Uj )} in L2 (0; T ; [W 1; 2 ()]∗ ). Therefore, we can assume that ∗ (Uj ) converges weakly in H 1 (0; T ; [W 1; 2 ()]∗ ) to some functional h0 . Using the strong convergence of {Uj } to UO 0 in L2 (QT ), it is simple to see that h0 = ∗ (UO 0 ). Now we are able to pass to the limit j → ∞ in (5.31) to get , ∗ O n @(UO 0 − G0 ) @’ @ (U 0 ) ∗ O ; ’ dt + (U 0 )bi t; x; @t @x @xi 0 Q, i=1 + a∗ (t; x; G0 ; UO 0 )’
d x dt = 0;
UO 0 (0; x) = h(x); x ∈ :
’ ∈ L2 (0; T ; W 1; 2 ()); (5.32)
Adapting the uniqueness result of Theorem 4 from [9], we obtain from (5.32) UO 0 =Ag0 and this ends the proof of Lemma 6. End of the proof of Theorem 5. We have had reduced the solvability of problem (1.1)– (1.4) to that of Eq. (5.16). The solvability of the last equation follows via Leray– Schauders’s principle from the above formulated statements (1), (2), which are consequences of Lemma 6. Therefore the proof of Theorem 5 is complete provided condition ($1 ) is satis8ed. In the case of condition ($2 ) the same arguments can be used. But it is not necessary to pass to the regularized problem (5.3)–(5.5) in that case.
6. Proof of the Uniqueness Theorem Assume by contradiction the existence of two solutions (uj ; vj ), j = 1; 2; of problem (1.1)–(1.4) in the sense of De8nition 1. We shall show that u1 (t; x) = u2 (t; x), v1 (t; x) = v2 (t; x). By Theorems 2–4 we have @uj @vj + 6 M13 (6.1) uj L∞ (QT ) + vj L∞ (QT ) + @x L2 (QT ) @x L2 (QT ) with a constant M13 depending only on known parameters. Let us now prove two auxiliary estimates.
H. Gajewski, I.V. Skrypnik / Nonlinear Analysis 56 (2004) 803 – 830
827
First auxiliary estimate: We have for almost all , ∈ (0; T ) @(u1 − u2 ) 2 2 d x dt |u1 (,; x) − u2 (,; x)| + @x Q, @(v1 − v2 ) 2 + |v1 − v2 |2 6 c55 @x Q, @v1 + @x
@u1 + 1 + $(t; x) |u1 − u2 |2 d x dt: @x
(6.2)
We shall obtain this estimate from equality (5.22) with (ui ), (ui ), a(t; x; vi , ui ), ui , vi instead of ∗ (Ui ), ∗ (Ui ), a∗ (t; x; Gi , Ui ), Ui , Gi respectively. Indeed, using (6.1) and the local Lipschitz conditions for resp. for a(t; x; ·; u), we get (u1 ) − (u2 ) − (u1 ) [(u1 ) − (u2 )] 6 c56 |u1 − u2 |2 (6.3) (u1 ) and
a(t; x; v1 ; u2 ) (u1 ) − (u2 ) − a(t; x; v2 ; u2 )(u1 − u2 ) (u1 ) (u1 ) − (u2 ) 6 |a(t; x; v1 ; u2 )| · − (u1 − u2 ) (u1 ) +|a(t; x; v1 ; u2 ) − a(t; x; v2 ; u2 )| · |u1 − u2 | 6 c57 {[1 + $(t; x)] |u1 − u2 |2 + |v1 − v2 |2 }:
(6.4)
Now (6.2) follows by using (6.3) and (6.4) in the same way as (5.29). Second auxiliary inequality: We have for almost all , ∈ (0; T )
2 @u2 2 2 2 @u1 d x dt |u1 − u2 | |u1 (,; x) − u2 (,; x)| d x + + @x @x Q, @(u1 − u2 ) 2 @(v1 − v2 ) 2 + 6 c58 @x @x Q,
@v1 2 @v2 2 2 2 |u1 − u2 | + [1 + $(t; x)] |u1 − u2 | d x dt: + + @x @x
(6.5)
The proof of inequality (6.5) coincides with that of inequality (6.8) in [7]. That proof is based on testing integral identity (2.8) for u = uj , v = vj with the test functions ’j given by 1 ’1 = [exp(N(u1 )) − exp(N(u2 ))]+ ; ’2 = N [u1 − u2 ]+ exp(N(u2 ))]; (u1 ) where N = max {|1 − 2 (s)|=2 (s): |s| 6 M13 } and M13 is the constant from (6.1). Remark that this proof is independent of the equation for the function v.
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We shall use also the estimate @(v1 − v2 ) 2 2 dx + |u1 − u2 |2 d x |v − v | d x 6 c 1 2 59 @x
(6.6)
following from Lemma 1 and (6.1). Now we can turn to the proof of Theorem 6. Applying Cauchy’s inequality to the term with |@v1 =@x| in (6.2), we obtain from (6.2), (6.5), (6.6) @(u1 − u2 ) 2 2 d x dt |u1 (,; x) − u2 (,; x)| + @x Q, @v1 2 @v2 2 |u1 − u2 |2 d x dt: 6 c60 1 + $(t; x) + + (6.7) @x @x Q, From condition (iii) and Theorem 3 we have @v1 2 @v2 2 p˜ + 1 + $(t; x) + @x ∈ L (QT ); @x
p˜ = min
p1 ;
p3 + 2 2
¿
n+2 : 2
Estimating the integral on the right-hand side of (6.7) by HPolder’s inequality, we get @v1 2 @v2 2 |u1 − u2 |2 d x dt 1 + $(t; x) + + @x @x Q, 6 c61
2p˜
|u1 − u2 |
Q,
1=p˜ d x dt
:
(6.8)
Applying HPolder’s and Young’s inequalities and the embedding V 2 (Q, ) → L2(n+2)=n (Q, ), where @u(t; x) 2 2 2 uV 2 (Q, ) := ess sup u (t; x) d x + @x dt d x t∈(0;,)
Q,
(cf. [14]), we can estimate the last integral in (6.8) as follows: 1=p˜ 2p˜ |u1 − u2 | d x dt Q,
@(u1 − u2 ) 2 d x dt |u1 (@; x) − u2 (@; x)| d x + 6 ' sup @x 0¡@¡, Q, |u1 − u2 |2 d x dt + c62 '−(n+2)(p˜ −1)=(n+2−np˜ )
2
Q,
(6.9)
with an arbitrary positive number '. With a suitable ' inequalities (6.7)–(6.9) imply |u1 (t; x) − u2 (t; x)|2 d x dt |u1 (,; x) − u2 (,; x)|2 d x 6 c63 (6.10)
Q,
for all , ∈ (0; T ). Gronwall’s lemma and the last estimate yield u1 = u2 . By (6.6) this implies v1 = v2 and the proof of Theorem 6 is complete.
H. Gajewski, I.V. Skrypnik / Nonlinear Analysis 56 (2004) 803 – 830
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Proof of Proposition 1. With the solution u of (1.1)–(1.4) we de8ne u1 (t; x) = u(t; x)
u2 (t; x) = u(t + 2; x);
2 ∈ (0; T − t)
and test the integral identity (2.8) with the functions ’i , i = 1; 2, given by t2 ’1 (t; x) = ∗ [∗ (u1 (t; x)) − ∗ (u2 (t; x))]; (u1 (t; x)) ’2 (t; x) = t 2 (u1 (t; x) − u2 (t; x)): Then, arguing essentially as in the proof of (6.7) and (6.10), we obtain @(u1 − u2 ) 2 d x dt ,2 |u1 (,; x) − u2 (,; x)|2 d x + t 2 @x Q, t|u1 − u2 |2 d x dt: 6 c64 Q,
(6.11)
Then, using the interpolation inequality c v2L2 () 6 'Sv2L2 () + v2[W 1; 2 ()]∗ : ' with v(x) = u1 (t; x) − u2 (t; x), ' = t' we can replace inequality (6.11) by 2 2 2 2 @ |u1 (,; x) − u2 (,; x)| d x + t (u1 − u2 ) d x dt , @x
6 c65
0
Q,
,
u1 (t; ·) − u2 (t; ·)2[W 1; 2 ()]∗ dt:
Since @u=@t ∈ L2 (0; T ; [W 1; 2 ()]∗ ), the di#erence quotient (1=2)[u(t + 2; x) − u(t; x)], is bounded in L2 (0; T ; [W 1; 2 ()]∗ ) uniformly with respect to 2. The last inequality now implies uniform boundedness of t [u(t + 2; x) − u(t; x)] 2 in L∞ (0; T ; L2 ()) ∩ L2 (0; T ; [W 1; 2 ()]∗ ) that implies the assertion of Proposition 1. References [1] G. Albinus, H. Gajewski, R. HPunlich, Thermodynamic design of energy models of semiconductor devices, Nonlinearity 15 (2002) 367–383. [2] H.W. Alt, S. Luckhaus, Quasilinear elliptic–parabolic di#erential equations, Math. Z. 183 (1983) 311–341. [3] Ph. Benilan, P. Wittbold, On mild and weak solutions of elliptic–parabolic systems, Adv. Di#erential Equations 1 (1996) 1053–1073. [4] C.M. Elliot, H. Garcke, On the Cahn–Hilliard equation with degenerate mobility, SIAM J. Math. Anal. 27 (1996) 404–423. [5] H. Gajewski, On a nonlocal model of nonisothermal phase separation, Adv. Math. Sci. Appl. 12 (2002) 569–586. [6] H. Gajewski, K. GrPoger, Reaction–di#usion processes of electrically charged species, Math. Nachr. 177 (1996) 109–130.
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[7] H. Gajewski, I.V. Skrypnik, On the uniqueness of solutions for nonlinear elliptic–parabolic equations, J. Evolution Equations 3 (2003) 247–281. [8] H. Gajewski, I.V. Skrypnik, To the uniqueness problem for nonlinear elliptic equations, Nonlinear Anal. 52 (2003) 291–304. [9] H. Gajewski, I.V. Skrypnik, On the uniqueness of solutions for nonlinear parabolic equations, Discrete Continuous Dyn. Systems 10 (2004) 215–236. [10] H. Gajewski, K. Zacharias, Global behavior of a reaction–di#usion system modelling chemotaxis, Math. Nachr. 195 (1998) 77–114. [11] H. Gajewski, K. Zacharias, On a nonlocal phase separation model, J. Math. Anal. Appl. 286 (2003) 11–31. [12] G. Giacomin, J.L. Lebowitz, Phase segregation dynamics in particle systems with long range interactions. I. Macroscopic limits, J. Statist. Phys. 87 (1–2) (1997) 37–61. [13] G. Giacomin, J.L. Lebowitz, Phase segregation dynamics in particle systems with long range interactions. II. Interface motion, SIAM J. Appl. Math. 58 (6) (1998) 1707–1729. [14] O.A. Ladyzhenskaya, V.A. Solonnikov, N.N. Uraltseva, Linear and quasilinear equations of parabolic type, in: Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence RI, 1974. [15] W. Littman, G. Stampacchia, H.F. Weinberger, Regular points for elliptic equations with discontinuous coeMcients, Ann. Scuola Norm. Sup. Pisa 17 (3) (1963) 43–77. [16] F. Otto, L1 –contraction and uniqueness for quasilinear elliptic–parabolic equations, C.R. Acad. Sci. Paris 318 (1) (1995) 1005–1010. [17] I.V. Skrypnik, Methods for analysis of nonlinear elliptic boundary value problems, in: Translations of Mathematical Monographs, Vol 139, American Mathematical Society, Providence, RI, 1994, xii+348pp.