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A model porous medium equation with variable exponent of nonlinearity: existence, uniqueness and localization properties of solutions
Nonlinear Analysis 60 (2005) 515 – 545 www.elsevier.com/locate/na
A model porous medium equation with variable exponent of nonlinearity: existence, uniqueness and localization properties of solutions S.N. Antontseva , S.I. Shmarevb,∗ a Departamento de Matemática, Universidade da Beira Interior, Rua Marqués d’Ávila e Bolama, 6201-001
Covilhã, Portugal b Departamento de Matemáticas, Universidad de Oviedo, c/Calvo Sotelo, s/n, 33007 Oviedo, Spain
Received 1 July 2004; accepted 14 September 2004
Abstract We study the localization properties of weak solutions to the Dirichlet problem for the degenerate parabolic equation ut − div (|u|(x,t) ∇u) = f, with variable exponent of nonlinearity . We prove the existence and uniqueness of weak solutions and establish conditions on the problem data and the exponent (x, t) sufficient for the existence of such properties as finite speed of propagation of disturbances, the waiting time effect, finite time vanishing of the solution. It is shown that the solution may instinct in a finite time even if ≡ (x) 0 in the problem domain but max = 0. 䉷 2004 Elsevier Ltd. All rights reserved. MSC: 35K55; 35K65; 35B05; 35B30 Keywords: Nonlinear parabolic equations; Nonstandard growth conditions
∗ Corresponding author. Departamento de Matemáticas, Universidad de Oviedo, c/Calvo Sotelo, s/n, 33007 Oviedo, Spain. Tel.: +34 98 510 31 95; fax: +34 98 510 33 54. E-mail addresses: [email protected] (S.N. Antontsev), [email protected] (S.I. Shmarev).
0362-546X/$ - see front matter 䉷 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2004.09.026
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1. Introduction The paper addresses the Dirichlet problem for a class of semilinear parabolic equations with variable exponents of nonlinearity. Let ⊂ RN be a domain with Lipschitz-continuous boundary *, and QT = × (0, T ] be a cylinder of the height T < ∞. We consider the following problem: ut − div (|u|(x,t) ∇u) = f
in QT ,
u = 0 on T = * × [0, T ],
u(x, 0) = u0 (x) in .
(1) (2)
Throughout the paper we assume that is a bounded function defined on QT such that ∀ (x, t) ∈ QT ,
−1 < − (x, t) + < ∞,
(3)
where − and + are given constants. There exists an abundant literature devoted to the study of the questions of existence, uniqueness and qualitative properties of solutions to nonlinear parabolic equations of type (1) with constant exponent of nonlinearity. We refer to the bibliography given in [5,11,13,17]. The solutions of equation (1) with = const = 0 possess properties which are not displayed by the solutions of the linear equation with = 0. It is known that if > 0, then the disturbances from the data propagate with finite speed, while for < 0 the solutions extinct in a finite time. These properties of solutions to Eq. (1) were first discovered in [25,27]. In the recent years, much attention is paid to the study of mathematical models of electrorheological fluids. These models include parabolic or elliptic equations nonlinear with respect to the gradient of the thought solution, and with variable exponents of nonlinearity—see [1,24] and the references therein. To the best of our knowledge, there are no results about the existence, uniqueness and qualitative properties of solutions to Eq. (1) with variable exponents . However, several authors studied the limit behaviour of solutions to Eq. (1) with constant exponents when → ∞ [12,14], and the behaviour of “almost linear” parabolic equations with singularly perturbed exponents [18–20]. Equations of type (1) appear in a natural way in continuum mechanics. Let us consider a motion of an ideal barotropic gas through a porous medium. Let be the gas density, V velocity and p pressure. The motion is governed by the mass conservation law
t + div (V ) = 0, the Darcy law which for the non-homogeneous medium has the form V = −k(x)∇ p,
k(x) is a given matrix
and the equation of state p = P (). It is usually assumed that P (s) = s with , = const. The above conditions then lead to the semi-linear parabolic equation for the density :
t =
div (k(x)∇ 1+ ). 1+
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517
If we additionally assume that p may explicitly depend on (x, t) and has the form p = (x,t) , the equation for becomes
t = div (k(x) ∇ (x,t) ) and can be written in the form
t = div (k(x) ∇ + ( ln ) k(x) · ∇ ).
(4)
In the present work, we limit ourselves to the study of Eq. (1) which is a simplified version of Eq. (4). The weak solution to problem (1)–(2) is understood as follows. Definition 1. A locally integrable bounded function u(x, t) is said to be weak solution of problem (1)–(2) if: (i) u ∈ L∞ (0, T ; ∞, ), |u|(x,t)/2 ∇u ∈ L2 (0, T ; L2 ()), (ii) u = 0 on * × (0, T ) in the sense of traces, (iii) for any test-function (x, t) satisfying the conditions
∈ L2 (0, T ; W01, 2 ()) ∩ L∞ (0, T ; L∞ ()),
ut ∈ L2 (0, T ; W0−1,2 ()),
t ∈ L2 (0, T ; L2 ())
and every 0 t1 t2 T the following integral identity holds: t2 t2 (x,t) (−ut + |u| ∇u∇ − f ) dx dt = − u dx . t1
(5)
t1
We prove first the existence of a weak solution to problem (1)–(2) in the sense of this definition. The uniqueness of weak solution is proved under the restriction (x, t) > 0. The study of the localization properties of weak solutions is based on the application of the local energy estimates. We establish the following properties: • Let u0 = 0 in Br (x0 ) = {x ∈ : |x − x0 | < r}. We say that the solution possesses the property of finite speed of propagation of disturbances from the data if there exist a function (t) > 0 and t∗ > 0 such that u(x, t) ≡ 0
in B(t) (x0 ) for all t ∈ [0, t∗ ].
We show that the weak solution of Eq. (1) possesses this property if (x, t) is continuous at the point (x0 , 0) and (x0 , 0) > 0. • Let u0 = 0 in a ball Br (x0 ) and assume that Br (x0 ) touches the set supp u0 at a point . We say that the solution u(x, t) has the waiting time property at the point if u(x, t) = 0 in Br (x0 ) for all t ∈ [0, t∗ ]. We establish conditions on the data u0 and f sufficient for the existence of this effect. • Let us say that the solution has the property of finite time extinction if there exists t∗ such that u = 0 in for all t t∗ . We show that this is true if (x, t) + < 0 in QT . We also derive estimates on the rate of vanishing of the norm of the weak solution when t → ∞.
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• It is shown that the condition (x, t) + < 0 is sufficient but not necessary for the finite time extinction. We prove that the solution way vanish in a finite time even if 0 but is not separated away from zero. The results of this paper were briefly announced in [10].
2. Existence of weak solution This section is devoted to the proof of existence of solutions to problem (1)–(2). We prove the following theorem. Theorem 1. Let (x, t) be a measurable in QT function, satisfying condition (3) and such that ∇ ∈ L2 (QT ). If u0 ∞, +
T 0
f (x, t)∞, dt = K(T ) < ∞,
(6)
then problem (1)–(2) has at least one weak solution in the sense of Definition 1. The solution is bounded and satisfies the estimate u∞,QT K(T ) with the constant K(T ) from condition (6). 2.1. Regularized problems Let us consider the auxiliary nonlinear parabolic problem
Lu ≡ ut − div (a( , u, M, x, t))∇u) = f u(x, 0) = u0 in , u = 0 on T .
in QT ,
(7)
Here M stands for a positive parameter to be chosen later. Let 0 < C ( , M, ± ) a = ( 2 + min{u2 , M 2 })(x,t)/2 C( , M, ± ).
(8)
The weak solution to problem (7) is constructed with the use of the Schauder Fixed Point Theorem [22, Chapter 4, Section 10]. Let us consider the linear problem
ut − div (( 2 + min{v 2 , M 2 })(x,t)/2 ∇u) = f u(x, 0) = u0 in , u = 0 on T ,
in QT ,
(9)
where ∈ [0, 1] is a real parameter, and v ∈ L2 (QT ) is a given function. Let BR = {v : v2,QT < R}. The solution u of problem (9) is considered as the solution of the functional equation u = (v) with
(u) : BR × [0, 1] → L2 (QT ).
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The existence of a solution of problem (7) in a ball BR = {v : v2,QT R} follows then from the following two facts: (1) the mapping (v) : BR → BR is continuous and compact, (2) for every ∈ [0, 1] the fixed points of the mapping u = (v) satisfy the estimate v2,QT R . Let us define the Banach space V as the completion of the space L2 (0, T ; W01,2 ()) in the norm vV = max v2, + ∇v2,QT [0,T ]
and denote by V0 the set of those elements of V for which 1 h
T −h
0
v(x, t + h) − v(x, t)22, dt → 0
when |h| → 0.
Compactness of the mapping (v) follows from the results on the solvability of linear parabolic equations with measurable coefficients [21, Chapter 3]: for every u0 ∈ L2 (), f, v ∈ L2 (QT ), measurable (x, t) satisfying condition (3), and any ∈ [0, 1] problem (9) has a unique solution u ∈ V0 . Moreover, by [21, Chapter 3, Theorem 4.5] the mapping
(v) is continuous in BR , for every test-function from the conditions of Definition 1
t2
t1
(−ut + a ∇u∇ − f ) dx dt = −
t2 u dx ,
(10)
t1
ut ∈ L2 (0, T ; W0−1,2 ()), uL∞ (QT ) C(R), and Eq. (9) is fulfilled in the sense of distributions. To check the fulfillment of the second condition for some R > 0 it amounts to derive the a priori estimates for all possible solutions of the nonlinear problem (9) with v = u and
∈ [0, 1]. The next subsection is entirely devoted to derive these estimates. 2.2. A priori estimates We consider the nonlinear problems: ut − div (a( , u, M, x, t)∇u) = f in QT , u(x, 0) = u0 in , u = 0 on T ,
(11)
with ∈ [0, 1] and the coefficient a( , u, M, x, t) defined in (8). Lemma 1. The solution of problem (11) satisfies the estimate u∞,QT
T 0
f ∞, (t) dt + u0 ∞, K(T ).
(12)
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Proof. Multiplying Eq. (11) by u2k−1 and integrating over , we arrive at the relation 1 d a|∇u|2 u2(k−1) dx + (2k − 1) u(·, t)2k 2k, 2k dt 2k−1 = u f dx, k = 1, . . . , ∞. (13)
By Hölder’s inequality 2k−1
u f dx u(·, t)2k−1 2k, f (·, t)2k, ,
whence
k = 1, 2, . . . ,
d (u(·, t)2k, ) + (2k − 1) a|∇u|2 u2(k−1) dx dt f (·, t) , k = 1, 2, ... . u(·, t)2k−1 2k, 2k,
u(·, t)2k−1 2k,
Simplifying and then integrating this relation in t, we obtain the following estimates for the solutions of problem (11): t u(·, t)2k,
f (·, t)2k, dt + u(·, 0)2k, , k = 1, 2, . . . . 0
Passing to the limit when k → ∞, we obtain (12).
Corollary 1. Choosing M > K(T ), we have min{u2 , M 2 } = u2
and
a = ( 2 + u2 )(x,t)/2 .
(14)
Corollary 2. In special case when u0 0 and f 0, the solution u(x, t) is nonnegative in QT . Set u− = min{u, 0} 0. Then u− (x, 0) = 0, u− |T = 0 and 1 d − a|∇u− |2 dx 0. u (·, t)2, + 2 dt It follows that for every t > 0 0 u− (·, t)2, u− (x, 0)2, = 0, whence the required assertion. Lemma 2. The solutions of problem (11) satisfy the estimates √ a ∇u2,QT C, with a constant C not depending on . Proof. It follows from (13) with k = 1 that for every t ∈ (0, T ), ∈ [0, 1] 1
2 2 2 u(·, t)2, + a |∇u| dx dt u0 2, +
|u| |f | dx dt, 2 2 Qt QT
(15)
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which gives √ a ∇ u22,QT C(K, T ).
521
Corollary 3. The solutions of problem (11) satisfy the estimates ( 2 + u2 )
+ /4
∇u2,QT C,
(16)
with a constant C not depending on . Proof. Since |u| K a.e. in QT , then
2 + u 2 2 + K 2
(x,t)
2 + u 2 2 + K2
+ ,
(17)
and the assertion immediately follows from Lemma 2.
Lemma 3. The solutions of problem (11) satisfy the estimates a ∇ u2,QT C, with a constant C not depending on . Moreover, if + 0, then ∇ u2,QT C. Proof. Let us consider the functions: u − (u) ≡ ( 2 + s 2 ) /2 ds, = 0 on T . 0
This function can be taken for the test function in the integral identity (10). Notice that I= u t dx dt QT − = u ut ( 2 + u2 ) /2 dx dt QT u * 2 2 − /2 = s( + s ) ds dx dt QT *t 0 =
u(x,T )
u0 (x)
− /2
s ( 2 + s 2 )
ds dx.
By virtue of Lemma 1 |I | M ≡ M(− , K). It follows from (10) with = (u) that a (u) |∇ u|2 dx dt M +
|f | (u) dx dt M + M1 |f | dx dt QT
with
QT
K
M1 = max = 2 0
− /2
s ( 2 + s 2 )
QT
ds.
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Using (17) in the form a M2 (K, − ) (u), we arrive at the inequality a 2 |∇ u|2 dx dt C(K, − ). QT
Corollary 4. The solutions of problem (11) satisfy the estimates ( 2 + u2 )
+ /2
+
|u| ∇ u2,QT C,
∇u2,QT C,
(18)
with a constant C not depending on . Lemma 4. The solution of problem (11) has the weak derivative ut ∈ L2 (0, T ; W0−1,2 ()). Proof. It amounts to show that for every smooth function ∈ L2 (0, T ; W01,2 ), such that (x, 0) = (x, T ) = 0, T u t dx dt M L (0,T ;W 1,2 ()) , I = 0
2
0
with a constant M independent of and u. Indeed: by virtue of identity (10) and the uniform estimate of Lemma 3 I (a ∇ u2,QT + C()f 2,QT ) ∇ 2,QT , where C() is the constant in the Poincare inequality ∀ ∈ W01,2 (),
2, C()∇ 2, .
Let us write Eq. (11) in the form ut = div G,
(19)
with G = a∇u + f ∗ ∈ L2 (QT ),
div f ∗ = f,
f ∗ |T = 0.
Eq. (19) is fulfilled in the sense of distributions. If ∇u ∈ L2 (QT ), which corresponds to the case + 0 (see Lemma 3), then according to [21, Chapter III, Lemma 4.1] the function u(x, t) satisfies the estimates
u(x + ei h, t) − u(x, t)2 h C|h|2 , i = 1, . . . , n, 2,QT (20) u(x, t + h) − u(x, t)2 h C|h|, 2,QT
where QhT = {(x, t) ∈ QT : (x + ei h, t), (x, t + h) ∈ QT , i = 1, . . . , n}, h is a scalar parameter, ei are the unit vectors of the axes xi .
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Lemma 5. Let u(x, t) be a solution of problem (11) and + 0. The function
u(x,t)
z(x, t) = z[u(x, t)] ≡
+
|s| ds
0
satisfies the estimates
z(x + ei h, t) − z(x, t)2 z(x, t + h) − z(x, t)2
2,QhT
2,QhT
C|h|2 ,
i = 1, . . . , n, (21)
C|h|.
+
Proof. Since ∇ z = |u| ∇ u, by virtue of (18) there holds the inclusion ∇ z ∈ L2 (QT ). The first of estimates (21) immediately follows then from the inequality |z(x + ei h, t) − z(x, t)|2 dx C, i = 1, . . . , n. |h|2 QhT To derive the second estimate we introduce the functions:
u = u(x, t + h) − u(x, t),
z = z(x, t + h) − z(x, t).
Then (z, u) = z,
z =
u1 u2
t+h t
+
|s| ds u,
∇ G(·, ) d
= − ∇(z),
t+h t
G(·, ) d
∇(z)L2 (Qh ) C.
,
(22)
(23)
T
Integrating equality (22) with respect to t ∈ (0, T − h), applying Hölder’s inequality and (23), we arrive at the inequality 2 z22,Qh T
T −h
0
C
T −h
0
t T −h
C |h| 0
T −h
|∇(z)|2 dx dt 0 t+h
t
2
|G(x, )| d
t+h
t+h
t
G(x, ) d
dx dt
dx dt
|G(x, )| d dx dt 2
C |h|.
2.3. Passage to the limit Let {u } be the sequence of solutions of problem (7). If + 0, the sequence {u } is compact by virtue of (20). If + 0, then according to (21) the sequence z[u (x, t)] is compact in Lq (QT ) with some 1 < q < ∞ [26]. The sequence {u } is also compact. Moreover, both sequences are bounded and converge a.e. in QT . The sequence {u } converges also a.e. in QT for −1 < − (x, t) + < ∞. Since u ∞,QT are uniformly bounded, we can
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extract a subsequence of the sequence {u } (which is assumed to coincide with the whole of the sequence) such that
u → u a.e. in QT , u in Lp (Q) with any 1 < p < ∞, → u strongly u dx → u dx for a.e. t ∈ (0, T ).
(24)
Consider the function g[u] = u( 2 + u2 )/2 ,
u2 + 1 ( 2 + u2 )/2 ∇ u 2 + u 2 1 + ∇ u ( 2 + u2 )/2 ln ( 2 + u2 ). 2
∇g =
If (x, t) + 1 > 0, then ∇ g ∈ L2 (QT ) because 2 2 u 1 2 2 2 2 (+1)/2 ) ln ( + u ) ∇ |∇ g|2 = ( + 1) 2 a ∇ u + + u ( 2 +u 2
C1 |a ∇ u|2 + C2 |∇ |2 , with constants Ci depending on ± and sup |u|. Then there exist functions u, g ∗ such that
g ≡ g[u ] = u ( 2 + u2 )/2 → u|u| a.e. in QT and strongly in Lq (QT ) with 1 < q < ∞, ∇g → ∇g ∗ weakly in L2 (QT ).
Let us identify the limit ∇ g ∗ . Notice that (1 + )u2 + 2 2 ( + u2 )/2 ∇ u 2 + u 2 1 = ∇ g − ∇ u ( 2 + u2 )/2 ln ( 2 + u2 ). 2
(25)
For every test-function lim ∇ g dx dt = →0 QT
∇ g ∗ dx dt QT = − lim g ∇ dx dt →0 QT = − u |u| ∇ dx dt, QT
whence ∇ g ∗ = ∇ (u|u| ) = (1 + )|u| ∇ u + 21 u|u| (ln u2 ) ∇
a.e. in QT .
(26)
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Given a test-function from the conditions of Definition 1, we can pass to the limit when → 0 in each term of the integral relation 5 i=1
( ) Ii
≡
−u t + ( 2 + u2 )(x,t)/2 ∇u ∇ − f dx dt
QT
+ By virtue of (24) ( ) I1 → −
QT
t=T u dx = 0. t=0
( ) I4
u t dx dt,
( ) + I5
→
t=T u dx t=0
when → 0.
Next, according to (25) and (26) 2 + u2 ( ) I2 = 2 2 QT (1 + )u +
1 2 2 /2 2 2 × ∇ g − [u ( + u ) ] ln ( + u ) ∇ ∇ dx dt 2
1 1 2 → ∇ (u|u| ) − u|u| (ln u ) ∇ ∇ dx dt +1 2 QT = |u| ∇ u ∇ dx dt when → 0. QT
Gathering these formulas we conclude that the limit function is a weak solution of problem (1)–(2). 3. Uniqueness of weak solution Theorem 2. Let the conditions of Theorem 1 be fulfilled and the function (x, t) be such that
(x, t) − > 0 in QT ,
sup |∇ | ∈ L2 (0, T ).
x∈
Then the weak solution of problem (1)–(2) is unique. Proof. In the proof we follow ideas of the papers [3,4,6–9]. Let us assume that there are two different solutions problem (1)–(2) u1 and u2 . Consider the function u = u1 − u2 . For every t ∈ (0, T ) the function u satisfies the integral identity t (−ut + (|u1 |(x,t) ∇u1 − |u2 |(x,t) ∇u2 )∇ ) dx dt 0 = − u(x, T )(x, T ) dx. (27)
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Consider the function W (x, t) ≡ W [u(x, t)] =
1 u |u|(x,t) , 1 + (x, t)
which possesses the following properties: ∇ 1+ ≡ |u| ∇ u + F (u, x, t),
∇ W (x, t) = |u| ∇ u + u |u|
ln u2 1 − 2 1+
|u1 | ∇ u1 − |u2 | ∇ u2 = (W [u1 ] − W [u2 ]) − (F (u1 , x, t) − F (u2 , x, t)). Then for every test-function :
(|u1 | ∇ u1 − |u2 | ∇ u2 ) ∇ dx dt = (∇ (W [u1 ] − W [u2 ]) ∇ + (F (u2 , x, t) − F (u1 , x, t)) ∇ ) dx dt.
QT
QT
Integrating by parts in the first term on the right-hand side, we arrive at the following representation:
(|u1 | ∇ u1 − |u2 | ∇ u2 ) ∇ dx dt = (u1 − u2 ) [−A + B ∇ ] dx dt,
QT
QT
where the vector-valued function B has the form B = A ∇ ln (1 + ) + D ∇ and 0A =
W [u1 ] − W [u2 ] 1 u1 |u1 | − u2 |u2 | = , u1 − u 2 1+ u1 − u 2
F (u1 , x, t) − F (u2 , x, t) u1 − u 2 ∇ u1 |u1 |(x,t) ln u21 − u2 |u2 |(x,t) ln u22 = − . 2(1 + ) u1 − u 2
D=
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It is easy to verify that 0 A C,
|D| C |∇ |,
D2 |∇ |2 (u1 |u1 |(x,t) ln u21 − u2 |u2 |(x,t) ln u22 )2 C|∇ |2 , = 4(1 + ) (u1 − u2 ) (u1 |u1 | − u2 |u2 | ) A D2 C, A
B2 C|∇ |2 , A
with a constant C depending on − and max |ui |. Hence, identity (27) takes on the form u(x, T )(x, T ) dx. [−t − A + B ∇ ] u dx dt = −
QT
(28)
Let (x, t) = (x, T − t), where (x, t) is the solution of the following parabolic problem:
t − (A + ) + B ∇ = h in QT , (29)
(x, 0) = 0, = 0 on T . In these conditions > 0 is an arbitrary small parameter and h is an arbitrary function. If follows then that for every h ∈ L2 (QT ) and > 0 problem (29) has a unique continuous weak solution such that t , Dij2 ∈ L2 (QT ). Instead of identity (28) we now have: u [h + ] dx dt = 0. (30) QT
We proceed to derive a priori estimates on the derivatives of . Let h ∈ C(0, T ; C01 ()). Multiplying Eq. (29) by and integrating over we obtain the equality 1 d (31) |∇ |2 dx + (A + )| |2 dx = I1 + I2 , 2 dt where
I1 =
B ∇ dx,
I2 = −
h dx =
∇h ∇ dx.
By Hölder’s inequality 1 |B|2 1 2 |I1 | (A + )| | dx + |∇ |2 dx, 2 2 (A + ) 1 1 |I2 | |∇ |2 dx + |∇h|2 dx. 2 2 Gathering (31)–(34) we have d |∇ |2 dx + (A + )| |2 dx dt
|B|2 |∇ h|2 dx + 1 |∇ |2 dx + (A + ) ≡ J.
(32)
(33) (34)
(35)
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By the above-established properties of A and B |B|2 |B|2 C sup(|∇ |)2 , (A + ) A
(36)
with a constant C not depending on . The right-hand side of (35) is estimated as follows: J (t)
|∇ | dx + 2
|∇ h|2 dx,
where (t) = C (1 + sup |∇ |2 ) with the same constant C. Plugging this estimate into (35) and applying Gronwall’s Lemma, we have |∇ |2 dx +
t 0
(A + )| |2 dx dt C
t 0
|∇h|2 dx dt.
(37)
Gathering (30) with (37) and applying Hölder’s inequality we have that
T
0
uh dx dt =
T
T
T
0
√ √ T meas
0
1/2
0
√ √ T meas
dx dt
| |2 dx dt 1/2
( + A)| |2 dx dt
→0
when → 0,
whence 0
T
uh dx dt = 0.
Since the last relation is true for any sufficiently smooth function h, it follows that u ≡ 0. Remark 1. We leave a gap between the conditions sufficient for the existence and uniqueness of weak solutions for problem (1)–(2). The existence is proved under the assumption −1 < (x, t) < ∞, while the uniqueness theorem is true if 0 < − (x, t) < ∞. It is known that the weak solution of the Cauchy problem for Eq. (1) with the constant exponent of nonlinearity (x, t) ≡ ∈ (−1, 0) need not be unique [23] (conditions of non-uniqueness of solutions in the limit case = −1 are given in [16]).
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4. Finite speed of propagation 4.1. Notation and assumptions Throughout the rest of the paper we use the following notation: given a point (x0 , t0 ) ∈ QT ≡ × (0, t0 ] B ≡ B (x0 ) = {x ∈ RN : |x − x0 | < } ⊂ , S ≡ S (x0 ) = *B (x0 ). We introduce the energy functions: b(, t) = ||u(·, t)||22,B ,
E(, t) =
t 0
E() = sup
0 t t0
b(, t),
u(x, ) |∇u|2 dx d , t E(, t) = 00 B u(x, ) |∇u|2 dx d ,
B
0 t t0
E + (, t) = t +
b() = sup
0
E () = sup
+
u |∇u|2 dx d , t + E + (, t) = 00 B u |∇u|2 dx dt ≡ E + (, t0 ).
B
0 t t0
We will consider the bounded weak solutions, which is why without loss of generality we always assume that |u| 1.
(38)
Since E(, t) is monotone nondecreasing in and t, there exist the weak derivatives t E (, t) = 0 S u(x, ) |∇u|2 dS d , Et (, t) = B u(x,t) |∇u|2 dx, Et (, t) = S u(x,t) |∇u|2 dS, E () = E (t , t0 ), + + E+ (, t) = 0 S u |∇u|2 dS dt, E () = E+ (, t0 ), + + + Et (, t) = B u |∇u|2 dx, E+t (, t) = S |u| |∇u|2 dS. It will be assumed that there is a 0 > 0 such that B0 (x0 ) ⊂ and b(, 0) = B (x0 ) u2 (x, 0) dx = 0, f (x, t) ≡ 0 in B0 (x0 ) × [0, T ]. 0
All the further considerations of this section are local and restricted to the cylinder G = B (x0 ) × {t ∈ [0, ]}
with = const 0 .
4.2. Finite speed of propagation of disturbances from the data Let us introduce the notion of local weak solution of Eq. (1).
(39)
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Definition 2. A locally integrable in G bounded function u(x, t) is said to be local weak solution of Eq. (1) in G if: (i) u ∈ L∞ (0, ; L∞ (B (x0 ))), |u|(x,t)/2 ∇u ∈ L2 (0, ; L2 (B (x0 ))), (ii) for any test-function (x, t) satisfying the conditions:
∈ L2 (0, ; W01,2 (B (x0 ))) ∩ L∞ (0, ; L∞ (B (x0 ))), t ∈ L2 (0, ; L2 (B(x0 )), the following integral identity holds: G
(−ut + |u|
(x,t)
∇u∇ − f ) dx dt = −
B (x0 )
u dx .
(40)
0
Obviously, every weak solution of problem (1)–(2) is a local weak solution of Eq. (1) in the sense of Definition 2. Theorem 3 (Finite speed of propagation). Let condition (39) be fulfilled,
(x0 , 0) > 0 and (x, t) ∈ C 0 (QT ). Then there exists 0 > 0 such that then every local weak solution of Eq. (1) in G0 satisfying (38) and the condition E0 = b(0 ) + E(0 ) < ∞ possesses the property of finite speed of propagation: u(x, t) = 0
for (x, t) ∈ B(t) (x0 ) × [0, 0 ],
where (t) given by the formula
(t) = 0 − Ct E0 ,
(41)
with positive constants , , , and C = C(b(0 ), 0 , ± , 0 , N ). Remark 2. Since (t) is a monotone decreasing function with (0) = 0 > 0, the set B(t) (x0 ) is nonempty for small t. The proof follows the ideas documented in the book [5] which is devoted to the study of localization properties of solutions to nonlinear equations with constant exponents of nonlinearity. We will use the material of this book as the analytical background for the further considerations, paying a special attention to the necessary changes in the arguments due to the presence of variable exponent of nonlinearity. The proof splits into three steps: (1) justification of the integration-by-parts formula for local weak solutions of Eq. (1); (2) derivation of a nonlinear ordinary differential inequality for the energy functions associated with the local weak solution; and (3) analysis of the ordinary differential inequalities.
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531
4.2.1. Formula of integration by parts. The energy relation Lemma 6. Let G = B (x0 ) × {t ∈ (0, ]} ⊂ QT . Under the conditions of Theorem 1, every local weak solution of Eq. (1) in G satisfies the energy relation =t t t 1 b(, ) + E(, t) + f u dx d = − u |u|(x,) |∇ u|2 dx d 2 0 B 0 S =0 (42) for any ∈ (0, ] and t ∈ (0, ]. Proof. Let u be a local weak solution of Eq. (1) in G ⊂ QT with some > 0. The fact that ∇ u need not belong to L2 (G ) prevents us from taking it as the test-function in the integral identity (40), which would immediately lead us to the desired energy relation. Let us denote u+ = max{u, 0} 0,
u− = max{−u, 0} 0,
u = u+ − u− .
Fix an arbitrary > 0, introduce the functions: ± u if u± > , = u± if u± − and then set u = u+ − u . For every > 0 the functions u possess the properties [21, Chapter 2, Section 4] ∇u if |u| > , ut if |u| > , u t = (43) ∇u = 0 if |u| < , 0 if |u| < .
Moreover, by virtue of (38) ∇ u ∈ L2 (G ) because |∇u |2 dx dt = |∇ u|2 dx dt ( ) G G + + − u(x,t) |∇ u|2 dx dt C − ,
(44)
G
( )
under the notation G = G ∩ {|u| > }. Next, G
u u t dx dt =
( )
G
u ut dx dt =
G
1 u u t dx dt = 2
u dx . 2
B (x0 )
(45)
0
Given > 0 and ∈ (0, ), we define the function n (r), r = |x − x0 |, by the formula if r ∈ 0, − n1 , 1 n (r) = n( − r) if r ∈ − n1 , , n ∈ N. 0 if r ∈ [, ],
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S.N. Antontsev, S.I. Shmarev / Nonlinear Analysis 60 (2005) 515 – 545
For every > 0 the function = u n (r) can be taken for the test-function in (5), which leads to the relation 1 2 n (x)u dx + ( ) (n |u| |∇u|2 + u |u| ∇u∇ n + f n u) dx dt − 2 B G 0 =− n (x) u u dx . (46) B 0
Letting → 0 we obtain 1 2
n u dx + (n (|u| |∇u|2 + f u)) dx dt = −Jn , 0 B 2
B
(47)
0
with Jn :=
B
0
u |u| ∇u∇ n dx dt.
Passing to the spherical coordinates with the origin at the point x0 , we have Jn =
=n =n
B
0
|u|(x,t) ∇u ∇ n dx dt
0
−(1/n)<|x−x0 |<
0
− n1
→ I (, )) =
u |u| ∇ u ·
x − x0 dx dt |x − x0 |
I (r) dr dt 0
S
u |u| ∇ u · n dS dt
when n → ∞
(n denotes the exterior unit normal to S (x0 )). Passing in (47) to the limit when n → ∞ we obtain the energy integral: 1 2
u2 dx + (|u|(x,t) |∇u|2 + f u) dx dt B 0 B 0 =− u |u|(x,t) ∇u · n dS dt ≡ −I (, ).
0
S
(48)
Theorem 4. Let under the conditions of Theorem 1 + 0. Then the weak solution of problem (1)–(2) satisfies the energy relation t t t 1 2 (x,t) 2 u dx + |u| |∇u| dx dt = f u dx dt. (49) 2 0 0 0
S.N. Antontsev, S.I. Shmarev / Nonlinear Analysis 60 (2005) 515 – 545
533
− Proof. Let {u+ } and {u } be the sequences of nonnegative functions defined by the relations
u+ = max{u; } − ,
u− = max{−u; } − .
The following formulas hold: ± ∇ u if ± u > , ± ∇ u = 0 if ± u < ,
u± ;t
=
± ut 0
if ± u > , if ± u < .
Moreover, since (x, t) + 0 and |u| 1, by virtue of (15) we have 2 −(x,t) (x,t) ± 2 dx dt |∇ u | |u| |∇ u± | dx dt = |u| QT QT = |u|−(x,t) (|u|(x,t) |∇ u|2 ) dx dt |u|/2 ∇ u22,QT , Q T
1,2 where Q T = {(x, t) ∈ QT : |u| > }. Thus, u± ∈ L2 (0, T ; W0 ()). Each of the functions u± can be taken now for the test-function in the Definition 1. The corresponding integral identities read as follows: t t 1 (x,t) ± 2 + u u± dx |u| |∇u | dx dt = ± f u± dx dt. 2 Q 0 0 t
Passing to the limit when → 0, for the functions u+ = max{u; 0} and u− = max{−u; 0} we obtain the energy relations t t 2 1 (u± )2 dx + |u± |(x,t) ∇u± dx dt = ± f u± dx dt. 2 Q 0 0 t Summing these relations and taking into account the formulas: u+ + u− = |u|,
u+ − u− = u,
u2 = (u+ )2 + (u− )2 ,
|∇ u|2 = |∇ u+ |2 + |∇ u− |2 ,
we obtain (49). 4.2.2. The ordinary differential inequality The energy relation (48) is valid for any t ∈ (0, T ) and every B (x0 ) ⊂ . If assumptions (39) are fulfilled, it can be specified as follows: for ∈ (0, 0 ], t ∈ [0, T ] t 1 2 u (x, t) dx + |u|(x,t) |∇u|2 dx dt = −I (, t), (50) 2 B (x0 ) 0 B (x0 ) with |I |
t 0
S (x0 )
|u|(x,)+1 |∇u| dS d ≡ .
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S.N. Antontsev, S.I. Shmarev / Nonlinear Analysis 60 (2005) 515 – 545 +
According to (38) |u|(x,t) |u| , and since t ∈ (0, T ) is arbitrary, we may write +
E(, t) 21 (E(, t) + E + (, t)) 21 (E() + E ()). Formula (50) can be now continued in the following way: +
b(, t) + E(, t) + E (, t) 6 .
(51)
We proceed to evaluate . Applying Hölder’s inequality and using (38) we have t 2 |u|(x,t)+2 dS dt E 0
E
t
sy (x0 )
−
S (x0 )
0
|u|2+ dS dt ≡ E 2 .
(52)
Let us make use of the trace-interpolation inequality [2]: for every fixed t ∈ (0, T ] + |u|2+ dS S
C
+
B
−
|u| |∇u| dx + 2
(2++ )/2
(b(, t))
+ )/2)(1−)
(b(, t))((2+
, (53)
with the exponents
=
N + = 2+ > 1. 2
N + + 2 ∈ (0, 1), N + + 4
In our notations, this inequality can be written as + + + |u|2+ dS C Et+ (, t) + − b(, t)(b(0 )) /2 (b(, t))((2+ )/2)(1−) S
+ C1 − Et+ (, t) + b(, t) (b(, t))((2+ )/2)(1−) ,
with the constant + C1 = C 0 + b(0 ) /2 . By Hölder’s inequality S (x0 )
|u|
2+−
dS
S (x0 )
with the exponent
= (N − 1)
+ − − 0. 2 + +
|u|
2++
(2+− )/(2++ ) dS
,
(54)
S.N. Antontsev, S.I. Shmarev / Nonlinear Analysis 60 (2005) 515 – 545
535
Gathering the above inequalities we obtain the estimate (2+− )/(2++ ) t0
2
+
S (x0 )
0
|u|2+ dS
dt
(2+− )/(2++ ) + C1 − [Et+ + b] b((2+ )/2)(1−) dt 0 t0 (2+− )/(2++ ) + dt C1 [Et+ + b] b((2+ )/2)(1−) C2 1− 0 t0 − + (2+− )(1−)/2 (Et+ + b)((2+ ))/(2+ ) dt C2 1− b t0
C3 1− t0 b
0 (2+− )(1−)/2
+
− )/(2++ ))
(E + b)((2+
+
+
≡ C3 1− t0 (E + b)2 C3 1− t0 (b + E + E )2 ,
(55)
with the constants (2+− )/(2++ )
C2 = C1
,
− )/2(2++ ))
C3 = C2 (1 + t0 )((2+
,
2 + − 2 + + + − − N + N + + 2 2 + − = − (N − 1) + 2+ , 2 + + 2 N + + 4 2 + + 2 + − 2 + − (1 − ) 2 + − , (1 − ) + + = = 4 2 2 + + 2 2 2 + + 2 + − 1 1− = > 0. 4 2 + +
= 1 − +
Let us claim that
> 0,
∈
1 ,1 2
in G0 .
(56)
Since (x, t) ∈ C 0 (QT ), (x0 , 0) > 0, and condition (56) is fulfilled for + = − = (x0 , 0), we may always choose 0 so small that (56) is fulfilled by continuity on the whole of G0 . Gathering (51), (52) and (55) and simplifying, we obtain the nonlinear ordinary differential inequality +
/2
(b() + E() + E ())1− C3 (1−)/2 t0 E . 1/2
(57) +
Keeping on the left-hand side only the term E and dropping the nonnegative terms E and b, we obtain the simplified inequality E
1−
C4 t0 1− E ,
with 0 < 1 − = 2(1 − ) < 1, C4 = C32 .
(58)
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S.N. Antontsev, S.I. Shmarev / Nonlinear Analysis 60 (2005) 515 – 545
4.2.3. Analysis of the differential inequality Integrating (58) in we have that for every ∈ (, 0 ):
0 E () E (0 ) − C5 t0− (0 − ), whence
E = sup
0 t t0
t0
E(, t) =
0
B
C5 =
, C4
(59)
u(x,t) |∇u|2 dx dt = 0,
if
(t) 0 −
t E (0 ), t ∈ [0, t0 ]. C5
(60)
By virtue of (57) we also have that b() = sup b(, t) = 0 0 t t0
and the conclusion follows: u(x, t) = 0
when x ∈ B(t) (x0 ), t ∈ [0, t0 ],
(61)
The proof of Theorem 3 is completed. 5. The waiting time effect Let us assume that in addition to condition (39) t 1/ ≡ b(, 0) + f 2 dx dt ( − 0 )+ , 0
B (x0 )
= const > 0
(62)
for ∈ [0 , 1 ] with the constant defined in the proof of Theorem 3. Theorem 5 (The waiting time property). Let the conditions of Theorem 3 be fulfilled and (62) be true. Then every local weak solution u(x, t) of Eq. (1) such that E1 =
sup
0 t 0
b(1 , 0 ) + E(1 , 0 ) < ∞
possesses the following property: there exist positive constants ∗ > 0 and t ∗ t0 T such that for every ∈ (0, ∗ ] u(x, t) = 0
in
B0 (x0 ) × [0, t ∗ ].
Proof. The energy relation (48) with = > 0 takes on the form t 1 u2 (x, t) dx + |u|(x,t) |∇u|2 dx dt = −I (, t) + J (, t), (63) 2 B (x0 ) 0 B (x0 )
S.N. Antontsev, S.I. Shmarev / Nonlinear Analysis 60 (2005) 515 – 545
with J (, t) =
1 2
B (x0 )
u2 (x, 0) dx −
t 0
B (x0 )
f u dx dt
537
(64)
and I (, t) defined in (48). Repeating the estimating performed in the proof of Theorem 3, we obtain the inequality +
+
/2
b() + E() + E () C3 −(−1)/2 t0 (E )1/2 (E + b) + |J |. The integral J is estimated in the following way: t f 2 dx dt + t0 b(), 2|J | b(, 0) + 0
(65)
t t0 .
B (x0 )
Assuming that t0 is small, from (65) we have that +
+
/2
b() + E() + E () C4 (−(−1)/2 t0 (E )1/2 (E + b) + ).
(66)
Applying Young’s inequality, we obtain +
/2
b() + E() + E () C5 ((−(−1)/2 t0 (E )1/2 )1/(1−) + ),
(67)
whence +
(b() + E() + E ())2(1−) C6 (−(−1) t0 E + 2(1−) ).
(68)
Letting 2(1− )=1− and using condition (62), we arrive at the nonhomogeneous ordinary differential inequality E
1−
(1−)/
2(1−) C(t0−1 − ( − 0 )+ 0 E +
).
Let t ∗ be chosen from the condition ∗ −1 A1− = C − 0 (t )
A + C 1− ,
with A > A0 = max{C 1− , E1 (1 − 0 )1/ }. It s easy to check that the function
E() ≡ A ( − 0 )1/ is the solution of the nonlinear ODE 1−
2(1−) E1− = C(t0−1 − ( − 0 )+ ), 0 E +
E(1 ) E1 .
The function E() is a majorant for E(), and the result follows: 1/
E() A( − 0 )+ ,
∈ [0 , 1 ].
(69)
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6. Asymptotic behaviour of solutions for large t Theorem 6. Let u(x, t) be a weak solution of problem (1)–(2) constructed in Theorem 1. Let
+ > 0.
(70)
If in Eq. (1) f (x, t) ≡ 0, then the solution u(x, t) satisfies the estimate y −1 (t)
−1
y0
−1
1 + t C0 ( − 1)y0
,
where y(t) = u(·, t)22, , y0 = y(0),
=1+
+ >1 2
and C0 is an absolute constant depending on + , , and N. If f ≡ / 0, there are positive constants K and B depending on , + and N such that if ||f (·, t)||2,
K (Bt + 1)(2−1)/2(−1)
,
then y(t)
y0 (Bt + 1)1/(−1)
.
Remark 3. If in the conditions of Theorem 1 −1 < − (x, t) + 0, we satisfy condition (70) choosing for + any positive number. Proof. Let u(x, t) be a weak solution of problem (1)–(2). For every t ∈ (0, T ) the energy relation (49) holds: t2 t2 t2 1 u2 dx + |u|(x,t) |∇u|2 dx dt = f u dx dt. (71) 2 t1 t1 t1 By the method of construction, every weak solution satisfies (|u|(x,t) |∇u|2 − f u) dx ∈ L1 [0, T ],
which allows us to differentiate (71) with respect to t: for a.e. t ∈ (0, T ) 1 d u2 (x, t) dx + |u(x, t)|(x,t) |∇u(x, t)|2 dx = − f u dx. 2 dt
(72)
S.N. Antontsev, S.I. Shmarev / Nonlinear Analysis 60 (2005) 515 – 545
539
Recall that the weak solutions are bounded. Not loosing generality we may assume therefore that |u| 1. Applying the imbedding theorem [15, Theorem 7.10], we have: for every 2N if N 2, 1 p N−2 ∞ if N = 1, there is a constant C ≡ C(p, N, + , ) such that 2/p + p(+ +2)/2 |u| dx C |∇(u( +2)/2 )|2 dx + |u|(x,t) |∇u|2 dx. =C |u| |∇u|2 dx C
Applying Hölder’s inequality we now have that
y(t) ≡
u2 dx C
C0
|u|
+ +2)/2
(x,t)
up(
|∇u| dx 2
4/p(2++ ) dx
2/(2++ )
,
(73)
with a new constant C0 = C0 (p, + , ). Substituting this inequality into (72), we have y + C0 y (t) , with
(74)
= f u dx ,
=
+ + 2 > 1. 2
(75)
Let f ≡ 0. Integrating inequality (74), we come to the following estimate on the rate of vanishing of the L2 ()-norm of the weak solution: y −1 (t)
−1
y0
−1
1 + tC 0 ( − 1)y0
→0
when t → ∞,
where y0 = u(x, 0)22, . Let f ≡ / 0. Applying Hölder’s inequality we estimate y(t) f (·, t)2, and arrive at the following nonhomogeneous ordinary differential inequality: √ (76) y + C0 y (t) yf (·, t)2, . √ Introduce the new thought function u(t) = y(t). This function satisfies the ordinary differential inequality u + 21 C0 u2−1 21 ||f (·, t)||2, .
(77)
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S.N. Antontsev, S.I. Shmarev / Nonlinear Analysis 60 (2005) 515 – 545
Let ||f (·, t)||2,
K (Bt + 1)(2−1)/2(−1)
,
(78)
with the constants (2−1)/2
K < C0 y0
,
(2−1)/2
B = (C0 y0
−1/2
− K)( − 1)y0
> 0.
(79)
Then the function √ w(t) = y0 (Bt + 1)−1/2(−1) solves the nonlinear ODE w + 21 Cw 2−1 (t) = 0,
w(0) =
√
y0
and is a majorant for the function u: u(t) w(t). Reverting to the function y(t) we get the estimate −1/(−1) |u(x, t)|2 dx y0 (Bt + 1) . (80) y(t) =
7. Vanishing of solutions in a finite time Theorem 7. Let the conditions of Theorem 1 be fulfilled and u(x, t) be a weak solution of problem (1)–(2). Let −1 < − (x, t) + < 0. If f (x, t) ≡ 0, then u(x, t) vanishes in the finite time: u(x, t) = 0
in for t t ∗ ,
where t ∗ depends only on u0 2, , + , and . Let f ≡ / 0. There exist > 0 and t > t ∗ such that if + +
t −(1+ )/ f (·, t)2, 1 − tf +
with tf t and 0 < < , Then u(x, t) ≡ 0
for t tf .
Proof. Arguing like in the previous section, for the function u2 (x, t) dt, y(t) =
(81)
S.N. Antontsev, S.I. Shmarev / Nonlinear Analysis 60 (2005) 515 – 545
541
we derive the ordinary differential inequality dy + Cy (t) , dt
(82)
with the right-hand side = f u dx ,
the exponent
+ <1 2 and a finite constant C which is defined through the constant in the embedding theorem. Let f ≡ 0. Integrating (82) we arrive at the estimate =1+
0 y 1− (t) y 1− (0) − C(1 − )t, whence y(t) ≡ 0
for t t ∗ =
y 1− (0) <∞ C(1 − )
(83)
and, consequently, u(x, t) = 0 for t t ∗ . For f ≡ / 0 the ordinary differential inequality for y(t) becomes √ y + Cy yf (·, t)2, . √ Introducing the function u(t) = y(t) we write the last inequality in the form u + 21 Cu (t) 21 ||f (·, t)||2, ,
= 2 − 1 = + + 1 < 1,
(84)
where f satisfies condition (81). Let us consider the ordinary differential equation + +
t −(1+ )/ 1 v + C v (t) = 1− . 2 2 tf +
(85)
The solutions to this ODE are majorants for u(t). It is easy to check that the function +
t −1/ u(t) = u(0) 1 − tf + satisfies (85) provided that the parameters u(0), ε, and tf are related by the condition −
1 1 ε 1 + + u(0) + Cu−(1+ )/ (0) − 0. 1− tf 2 2
Given u(0), we satisfy this condition increasing tf and decreasing .
(86)
Corollary 5. It follows from the proof that the presence of the effect of finite time vanishing of solutions is guaranteed by fulfillment of condition (86) which connects the three
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S.N. Antontsev, S.I. Shmarev / Nonlinear Analysis 60 (2005) 515 – 545
parameters: —the “intensity” of the right-hand side in Eq. (1), tf —the moment when the right-hand side vanishes, and u0 2, . The assertion of Theorem 7 remains true if we fix any of the three parameters and vary the other two in order to satisfy condition (86).
8. A limit case in the finite time vanishing: + = 0 The condition + < 0 was crucial for the proof of Theorem 7. However, this condition is sufficient but not necessary for the solution to extinct in finite time. A solution of problem (1)–(2) may extinct in finite time even if + = sup (x, t) = 0. Let us consider the following problem:
ut − (u(x) ux )x = 0 in QT = (0, 1) × (0, T ), u(0, t) = u(1, t) = 0, u(x, 0) = u0 (x) 0,
(87)
where the exponent ≡ (x) satisfies the conditions
−1 < − (x) + = 0 for x ∈ [0, 1], (x0 ) = 0 with either x0 = 0 or x0 = 1, ( (x))2 20 > 0, (x) 0.
(88)
By Theorem 1, under these conditions on (x) problem (87) has a nonnegative weak solution u(x, t) for every nonnegative u0 ∈ L2 (0, 1). Theorem 8. Let conditions (88) be fulfilled, u0 ∈ L2 (0, 1), and the condition 0 u0 1 holds. Then any nonnegative weak solution u(x, t) of problem (87) vanishes in the finite time: u(x, t) ≡ 0 for x ∈ [0, 1] and t t∗ = −
1 8 . 2 0 ln y(0)
Proof. The nonnegative weak solution of problem (87) satisfies the energy relation (72) 1 d 2 dt
1
1
u2 (x, t) dx +
0
0
u(x, t)(x) u2x (x, t) dx = 0,
(89)
whence y(t) ≡
1
u2 (x, t) dx y(0) < 1.
0
Moreover, 0 u(x, t) 1 in QT . Let us introduce the new function
u1+(x)/2
(x, t) ≡ [u(x, t)] = 0
ln d 0,
(0, t) = (1, t) = 0.
(90)
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543
A direct calculation gives:
2 ( )2 2+ 2 u ln u, u ux = (u1+/2 )2x − u1+/2 (u1+/2 )x ln u + 4 2 1+/2 x = 1 + )x u1+/2 ln u. (u 2 Gathering these two formulas we have that 1 ( )2 2+ 2 2 1+/2 2 u ux = (u )x + ln u − . u 4 1 + /2 x (1 + /2)2
1+
Since the function (x) is convex and 0, we also have that 1 1 x dx = − dx 0. (1 + /2)3 (1 + /2)3 0 0 Writing the energy relation in the form 1 1 ( )2 2+ 2 1 d 1 2 1+/2 2 (u u u (x, t) dx + )x + ln u dx 2 4 2 dt 0 0 (1 + /2) 1 + dx = 0. (1 + /2)3 0 and then dropping the nonnegative terms, we arrive at the inequality 1 1 ( )2 2+ 2 1 d 1 2 u (x, t) dx + ln u dx 0. u 2 2 dt 0 4 0 (1 + /2) In our assumptions on (x) and u(x, t), this inequality can continued as follows: 20 1 2 2 2 1 d 1 2 u (x, t) dx + u ln u dx 0. 2 dt 0 16 0 Let us consider now the function (s) ≡ s ln2 s. Since 2 (s) = (ln s + 1), s this function is convex on the interval s ∈ (0, 1/e). By Jensen’s inequality 1 1 1 u2 dx ≡ u2 dx ln2 u2 dx 0 0 0 1 1 (u2 ) dx ≡ u2 ln2 u2 dx
(s) = ln2 s + 2 ln s,
0
0
and (91) becomes 1 1 20 1 d 1 2 2 2 2 u (x, t) dx + u dx ln u dx 0. 2 dt 0 16 0 0
(91)
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S.N. Antontsev, S.I. Shmarev / Nonlinear Analysis 60 (2005) 515 – 545
Thus, the function y(t) = u(·, t)22,(0,1) satisfies the ordinary differential inequality
20 y ln2 y 0, 8
y +
y(0) < 1.
(92)
Any nonnegative solution of the ODE z +
20 z ln2 z = 0, 8
z(0) = y(0) < 1,
is a majorant for the function y(t). The function z(t) can be constructed explicitly: 1 . z(t) = exp 20 1 + t ln y(0) 8 This is a monotone decreasing function that vanishes at the instant t∗ = −
1 8 ∈ (0, ∞). 2 0 ln y(0)
Since z(t) y(t) 0, the assertion of theorem follows.
Remark 4. It is easy to see that unless we impose some additional assumptions on the data, the assertion of Theorem 8 is false if the function (x) ≡ 0 on an interval I =[a, b] ⊂ [0, 1]. Indeed, let us assume the contrary: (x) ∈ (−1, 0], (x) ≡ 0 for x ∈ [a, b] ⊂ [0, 1], and u(x, t) ≡ 0 for all x ∈ [0, 1], t ∈ [t∗ , T ]. We may consider u(x, t) as a nonnegative solution of the Dirichlet problem for the heat equation in the rectangle (a, b) × (0, t∗ ), so that the assumption u(x, t∗ ) ≡ 0 on [a, b] contradicts the strong maximum principle. Acknowledgements The first author was partially supported by the Project DECONT—Differential Equations of Continuum Mechanics, FCT/MCES (Portugal) at the “Centro de Matemática”, Universidade da Beira Interior, and by the Project FCT-POCTI /MAT/34471/2000. The second author was supported by the Spanish Research Project BFM2000-1324 and by the European RTN Programme HPRN-CT-2002-00274. References [1] A. Acerbi, G. Mignione, G. Seregin, Regularity results for a class of parabolic systems related to a class of non-newtonian fluids, Ann. Inst. H. Poincaré 21 (6) (2004) 25–60. [2] S.N. Antontsev, Localization of solutions of degenerate equations of continuum mechanics, Akademiya Nauk SSSR, Sibirsk. Otdel., Institut Gidrodinamiki, Novosibirsk, 1986 (in Russian). [3] S.N. Antontsev, J.I. Díaz, A.V. Domanski˘ı, Stability and stabilization of generalized solutions of degenerate problems of two-phase filtration, Dokl. Akad. Nauk 325 (1992) 1151–1155 (English Trans. in Soviet Phys. Dokl. 37(8) (1992) 411–413).
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[4] S.N. Antontsev, J.I. Díaz, A.V. Domanski˘ı, Continuous dependence and stabilization of solutions of the degenerate system in two-phase filtration, Dinamika Sploshn. Sredy, Fluids Dynamics Free Boundary 107 (1993) 11–25. [5] S.N. Antontsev, J.I. Díaz, S. Shmarev, Energy methods for free boundary problems: applications to nonlinear PDEs and fluid mechanics, Progress in Nonlinear Differential Equations and their Applications, vol. 48, Bikhäuser, Boston, 2002. [6] S.N. Antontsev, A.V. Domanski˘ı, Uniqueness of generalized solutions of degenerate problems of two-phase filtration, Chisl. Metody Mekh. Sploshn. Sredy 15(6) (1984) 15–27 (in Russian). [7] S.N. Antontsev, A.V. Domanskii, V.I. Penkovskii, Filtration in by-well zone of the formation and the well productivity stimulation problems, Lavrentyev Institute of Hydrodynamics, Siberian Branch of the USSR Academy of Science, Novosibirsk, 1989 (in Russian). [8] S.N. Antontsev, G. Gagneux, Petits paramètres et passages à la limite dans les problèmes de filtration diphasique, Progress Partial Differential Equations (Pont-à- Mousson 1997) 1 (1998). [9] S.N. Antontsev, A.V. Kazhikhov, V.N. Monakhov, Boundary Value Problems in Mechanics of Nonhomogeneous Fluids, North-Holland Publishing Co.,Amsterdam, 1990 (Translated from original Russian edition: Nauka, Novosibirsk, 1983). [10] S.N. Antontsev, S.I. Shmarev, A generalized porous medium equation with variable exponents of nonlinearity: existence, uniqueness and localization properties of solutions, The Seminar Dedicated to the Memory of I. Petrovskii, Book of Abstracts, Moskow State University, May 16–22, 2004. [11] D.G. Aronson, The porous medium equation, in: Nonlinear Diffusion Problems (Montecatini Terme, 1985), Lecture Notes in Mathematics, vol. 1224, Springer, Berlin, 1986, pp. 1–46. [12] P. Bénilan, L. Boccardo, M.A. Herrero, On the limit of solutions of ut = um as m → ∞, Rend. Sem. Mat. Univ. Politec. Torino (1991) 1–13 (Some topics in nonlinear PDEs, Turin, 1989). [13] E. Chasseigne, J.L. Vazquez, Theory of extended solutions for fast-diffusion equations in optimal classes of data. Radiation from singularities, Arch. Ration. Mech. Anal. 164 (2002) 133–187. [14] C.M. Elliott, M.A. Herrero, J.R. King, J.R. Ockendon, The mesa problem: diffusion patterns for ut = ∇ · (um ∇u) as m → +∞, IMA J. Appl. Math. 37 (1986) 147–154. [15] D. Gilbarg, N.S. Trudinger, Elliptic partial differential equations of second order, Grundlehren der Mathematischen Wissenschaften, vol. 224, Springer, Berlin, 1977. [16] S.-Y. Hsu, Removable singularities and non-uniqueness of solutions of a singular diffusion equation, Math. Ann. 325 (2003) 665–693. [17] A.S. Kalashnikov, Some problems of the qualitative theory of second-order nonlinear degenerate parabolic equations, Uspekhi Mat. Nauk 42 (1987) 135–176, 287. [18] A.S. Kalashnikov, Nonlinear effects in heat propagation in media with sources and sinks that are close to linear. Differentsial’nye Uravneniya 31 (2) (1995) 277–288, 366. [19] A.S. Kalashnikov, On some problems in the nonlinear theory of heat conduction with data containing a small parameter in the exponents, Zh. Vychisl. Mat. i Mat. Fiz. 35 (7) (1995) 1077–1094. [20] A.S. Kalashnikov, On some nonlinear problems in mathematical physics with exponents that are close to critical, Trudy Sem imeni I. G. Petrovskogo 19 (1996) 73–98, 346. [21] O.A. Ladyženskaja, V.A. Solonnikov, N.N. Ural’tseva, Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs, vol. 23, American Mathematical Society, Providence, RI, 1967 (Translated from the Russian by S. Smith). [22] O.A. Ladyženskaja, N.N. Ural‘ceva, Linear and Quasilinear Elliptic Equations, Academic Press, New York, 1968. (Translated from the Russian by Scripta Technica, Inc.). [23] A. Rodríguez, J.L. Vázquez, Non-uniqueness of solutions of nonlinear heat equations of fast diffusion type, Ann. Inst. H. Poincaré Anal. Non Linéaire 12 (1995) 173–200. [24] M. R˚užiˇcka, Electrorheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Mathematics, vol. 1748, Springer, Berlin, 2000. [25] E.S. Sabinina, On a class of non-linear degenerate parabolic equations, Dokl. Akad. Nauk SSSR 143 (1962) 794–797. [26] J. Simon, Compact sets in the space Lp (0, T ; B), Ann. Mat. Pura Appl. 146 (4) (1987) 65–96. [27] Y.B. Zel‘doviˇc, A.S. Kompaneec, On the theory of propagation of heat with the heat conductivity depending upon the temperature, In: Collection in Honor of the Seventieth Birthday of Academician A. F. Ioffe, Izdat. Akad. Nauk SSSR, Moscow, 1950, 61–71.
A model porous medium equation with variable exponent of nonlinearity: existence, uniqueness and localization properties of solutions S.N. Antontseva , S.I. Shmarevb,∗ a Departamento de Matemática, Universidade da Beira Interior, Rua Marqués d’Ávila e Bolama, 6201-001
Covilhã, Portugal b Departamento de Matemáticas, Universidad de Oviedo, c/Calvo Sotelo, s/n, 33007 Oviedo, Spain
Received 1 July 2004; accepted 14 September 2004
Abstract We study the localization properties of weak solutions to the Dirichlet problem for the degenerate parabolic equation ut − div (|u|(x,t) ∇u) = f, with variable exponent of nonlinearity . We prove the existence and uniqueness of weak solutions and establish conditions on the problem data and the exponent (x, t) sufficient for the existence of such properties as finite speed of propagation of disturbances, the waiting time effect, finite time vanishing of the solution. It is shown that the solution may instinct in a finite time even if ≡ (x) 0 in the problem domain but max = 0. 䉷 2004 Elsevier Ltd. All rights reserved. MSC: 35K55; 35K65; 35B05; 35B30 Keywords: Nonlinear parabolic equations; Nonstandard growth conditions
∗ Corresponding author. Departamento de Matemáticas, Universidad de Oviedo, c/Calvo Sotelo, s/n, 33007 Oviedo, Spain. Tel.: +34 98 510 31 95; fax: +34 98 510 33 54. E-mail addresses: [email protected] (S.N. Antontsev), [email protected] (S.I. Shmarev).
0362-546X/$ - see front matter 䉷 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2004.09.026
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1. Introduction The paper addresses the Dirichlet problem for a class of semilinear parabolic equations with variable exponents of nonlinearity. Let ⊂ RN be a domain with Lipschitz-continuous boundary *, and QT = × (0, T ] be a cylinder of the height T < ∞. We consider the following problem: ut − div (|u|(x,t) ∇u) = f
in QT ,
u = 0 on T = * × [0, T ],
u(x, 0) = u0 (x) in .
(1) (2)
Throughout the paper we assume that is a bounded function defined on QT such that ∀ (x, t) ∈ QT ,
−1 < − (x, t) + < ∞,
(3)
where − and + are given constants. There exists an abundant literature devoted to the study of the questions of existence, uniqueness and qualitative properties of solutions to nonlinear parabolic equations of type (1) with constant exponent of nonlinearity. We refer to the bibliography given in [5,11,13,17]. The solutions of equation (1) with = const = 0 possess properties which are not displayed by the solutions of the linear equation with = 0. It is known that if > 0, then the disturbances from the data propagate with finite speed, while for < 0 the solutions extinct in a finite time. These properties of solutions to Eq. (1) were first discovered in [25,27]. In the recent years, much attention is paid to the study of mathematical models of electrorheological fluids. These models include parabolic or elliptic equations nonlinear with respect to the gradient of the thought solution, and with variable exponents of nonlinearity—see [1,24] and the references therein. To the best of our knowledge, there are no results about the existence, uniqueness and qualitative properties of solutions to Eq. (1) with variable exponents . However, several authors studied the limit behaviour of solutions to Eq. (1) with constant exponents when → ∞ [12,14], and the behaviour of “almost linear” parabolic equations with singularly perturbed exponents [18–20]. Equations of type (1) appear in a natural way in continuum mechanics. Let us consider a motion of an ideal barotropic gas through a porous medium. Let be the gas density, V velocity and p pressure. The motion is governed by the mass conservation law
t + div (V ) = 0, the Darcy law which for the non-homogeneous medium has the form V = −k(x)∇ p,
k(x) is a given matrix
and the equation of state p = P (). It is usually assumed that P (s) = s with , = const. The above conditions then lead to the semi-linear parabolic equation for the density :
t =
div (k(x)∇ 1+ ). 1+
S.N. Antontsev, S.I. Shmarev / Nonlinear Analysis 60 (2005) 515 – 545
517
If we additionally assume that p may explicitly depend on (x, t) and has the form p = (x,t) , the equation for becomes
t = div (k(x) ∇ (x,t) ) and can be written in the form
t = div (k(x) ∇ + ( ln ) k(x) · ∇ ).
(4)
In the present work, we limit ourselves to the study of Eq. (1) which is a simplified version of Eq. (4). The weak solution to problem (1)–(2) is understood as follows. Definition 1. A locally integrable bounded function u(x, t) is said to be weak solution of problem (1)–(2) if: (i) u ∈ L∞ (0, T ; ∞, ), |u|(x,t)/2 ∇u ∈ L2 (0, T ; L2 ()), (ii) u = 0 on * × (0, T ) in the sense of traces, (iii) for any test-function (x, t) satisfying the conditions
∈ L2 (0, T ; W01, 2 ()) ∩ L∞ (0, T ; L∞ ()),
ut ∈ L2 (0, T ; W0−1,2 ()),
t ∈ L2 (0, T ; L2 ())
and every 0 t1 t2 T the following integral identity holds: t2 t2 (x,t) (−ut + |u| ∇u∇ − f ) dx dt = − u dx . t1
(5)
t1
We prove first the existence of a weak solution to problem (1)–(2) in the sense of this definition. The uniqueness of weak solution is proved under the restriction (x, t) > 0. The study of the localization properties of weak solutions is based on the application of the local energy estimates. We establish the following properties: • Let u0 = 0 in Br (x0 ) = {x ∈ : |x − x0 | < r}. We say that the solution possesses the property of finite speed of propagation of disturbances from the data if there exist a function (t) > 0 and t∗ > 0 such that u(x, t) ≡ 0
in B(t) (x0 ) for all t ∈ [0, t∗ ].
We show that the weak solution of Eq. (1) possesses this property if (x, t) is continuous at the point (x0 , 0) and (x0 , 0) > 0. • Let u0 = 0 in a ball Br (x0 ) and assume that Br (x0 ) touches the set supp u0 at a point . We say that the solution u(x, t) has the waiting time property at the point if u(x, t) = 0 in Br (x0 ) for all t ∈ [0, t∗ ]. We establish conditions on the data u0 and f sufficient for the existence of this effect. • Let us say that the solution has the property of finite time extinction if there exists t∗ such that u = 0 in for all t t∗ . We show that this is true if (x, t) + < 0 in QT . We also derive estimates on the rate of vanishing of the norm of the weak solution when t → ∞.
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• It is shown that the condition (x, t) + < 0 is sufficient but not necessary for the finite time extinction. We prove that the solution way vanish in a finite time even if 0 but is not separated away from zero. The results of this paper were briefly announced in [10].
2. Existence of weak solution This section is devoted to the proof of existence of solutions to problem (1)–(2). We prove the following theorem. Theorem 1. Let (x, t) be a measurable in QT function, satisfying condition (3) and such that ∇ ∈ L2 (QT ). If u0 ∞, +
T 0
f (x, t)∞, dt = K(T ) < ∞,
(6)
then problem (1)–(2) has at least one weak solution in the sense of Definition 1. The solution is bounded and satisfies the estimate u∞,QT K(T ) with the constant K(T ) from condition (6). 2.1. Regularized problems Let us consider the auxiliary nonlinear parabolic problem
Lu ≡ ut − div (a( , u, M, x, t))∇u) = f u(x, 0) = u0 in , u = 0 on T .
in QT ,
(7)
Here M stands for a positive parameter to be chosen later. Let 0 < C ( , M, ± ) a = ( 2 + min{u2 , M 2 })(x,t)/2 C( , M, ± ).
(8)
The weak solution to problem (7) is constructed with the use of the Schauder Fixed Point Theorem [22, Chapter 4, Section 10]. Let us consider the linear problem
ut − div (( 2 + min{v 2 , M 2 })(x,t)/2 ∇u) = f u(x, 0) = u0 in , u = 0 on T ,
in QT ,
(9)
where ∈ [0, 1] is a real parameter, and v ∈ L2 (QT ) is a given function. Let BR = {v : v2,QT < R}. The solution u of problem (9) is considered as the solution of the functional equation u = (v) with
(u) : BR × [0, 1] → L2 (QT ).
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519
The existence of a solution of problem (7) in a ball BR = {v : v2,QT R} follows then from the following two facts: (1) the mapping (v) : BR → BR is continuous and compact, (2) for every ∈ [0, 1] the fixed points of the mapping u = (v) satisfy the estimate v2,QT R . Let us define the Banach space V as the completion of the space L2 (0, T ; W01,2 ()) in the norm vV = max v2, + ∇v2,QT [0,T ]
and denote by V0 the set of those elements of V for which 1 h
T −h
0
v(x, t + h) − v(x, t)22, dt → 0
when |h| → 0.
Compactness of the mapping (v) follows from the results on the solvability of linear parabolic equations with measurable coefficients [21, Chapter 3]: for every u0 ∈ L2 (), f, v ∈ L2 (QT ), measurable (x, t) satisfying condition (3), and any ∈ [0, 1] problem (9) has a unique solution u ∈ V0 . Moreover, by [21, Chapter 3, Theorem 4.5] the mapping
(v) is continuous in BR , for every test-function from the conditions of Definition 1
t2
t1
(−ut + a ∇u∇ − f ) dx dt = −
t2 u dx ,
(10)
t1
ut ∈ L2 (0, T ; W0−1,2 ()), uL∞ (QT ) C(R), and Eq. (9) is fulfilled in the sense of distributions. To check the fulfillment of the second condition for some R > 0 it amounts to derive the a priori estimates for all possible solutions of the nonlinear problem (9) with v = u and
∈ [0, 1]. The next subsection is entirely devoted to derive these estimates. 2.2. A priori estimates We consider the nonlinear problems: ut − div (a( , u, M, x, t)∇u) = f in QT , u(x, 0) = u0 in , u = 0 on T ,
(11)
with ∈ [0, 1] and the coefficient a( , u, M, x, t) defined in (8). Lemma 1. The solution of problem (11) satisfies the estimate u∞,QT
T 0
f ∞, (t) dt + u0 ∞, K(T ).
(12)
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Proof. Multiplying Eq. (11) by u2k−1 and integrating over , we arrive at the relation 1 d a|∇u|2 u2(k−1) dx + (2k − 1) u(·, t)2k 2k, 2k dt 2k−1 = u f dx, k = 1, . . . , ∞. (13)
By Hölder’s inequality 2k−1
u f dx u(·, t)2k−1 2k, f (·, t)2k, ,
whence
k = 1, 2, . . . ,
d (u(·, t)2k, ) + (2k − 1) a|∇u|2 u2(k−1) dx dt f (·, t) , k = 1, 2, ... . u(·, t)2k−1 2k, 2k,
u(·, t)2k−1 2k,
Simplifying and then integrating this relation in t, we obtain the following estimates for the solutions of problem (11): t u(·, t)2k,
f (·, t)2k, dt + u(·, 0)2k, , k = 1, 2, . . . . 0
Passing to the limit when k → ∞, we obtain (12).
Corollary 1. Choosing M > K(T ), we have min{u2 , M 2 } = u2
and
a = ( 2 + u2 )(x,t)/2 .
(14)
Corollary 2. In special case when u0 0 and f 0, the solution u(x, t) is nonnegative in QT . Set u− = min{u, 0} 0. Then u− (x, 0) = 0, u− |T = 0 and 1 d − a|∇u− |2 dx 0. u (·, t)2, + 2 dt It follows that for every t > 0 0 u− (·, t)2, u− (x, 0)2, = 0, whence the required assertion. Lemma 2. The solutions of problem (11) satisfy the estimates √ a ∇u2,QT C, with a constant C not depending on . Proof. It follows from (13) with k = 1 that for every t ∈ (0, T ), ∈ [0, 1] 1
2 2 2 u(·, t)2, + a |∇u| dx dt u0 2, +
|u| |f | dx dt, 2 2 Qt QT
(15)
S.N. Antontsev, S.I. Shmarev / Nonlinear Analysis 60 (2005) 515 – 545
which gives √ a ∇ u22,QT C(K, T ).
521
Corollary 3. The solutions of problem (11) satisfy the estimates ( 2 + u2 )
+ /4
∇u2,QT C,
(16)
with a constant C not depending on . Proof. Since |u| K a.e. in QT , then
2 + u 2 2 + K 2
(x,t)
2 + u 2 2 + K2
+ ,
(17)
and the assertion immediately follows from Lemma 2.
Lemma 3. The solutions of problem (11) satisfy the estimates a ∇ u2,QT C, with a constant C not depending on . Moreover, if + 0, then ∇ u2,QT C. Proof. Let us consider the functions: u − (u) ≡ ( 2 + s 2 ) /2 ds, = 0 on T . 0
This function can be taken for the test function in the integral identity (10). Notice that I= u t dx dt QT − = u ut ( 2 + u2 ) /2 dx dt QT u * 2 2 − /2 = s( + s ) ds dx dt QT *t 0 =
u(x,T )
u0 (x)
− /2
s ( 2 + s 2 )
ds dx.
By virtue of Lemma 1 |I | M ≡ M(− , K). It follows from (10) with = (u) that a (u) |∇ u|2 dx dt M +
|f | (u) dx dt M + M1 |f | dx dt QT
with
QT
K
M1 = max = 2 0
− /2
s ( 2 + s 2 )
QT
ds.
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Using (17) in the form a M2 (K, − ) (u), we arrive at the inequality a 2 |∇ u|2 dx dt C(K, − ). QT
Corollary 4. The solutions of problem (11) satisfy the estimates ( 2 + u2 )
+ /2
+
|u| ∇ u2,QT C,
∇u2,QT C,
(18)
with a constant C not depending on . Lemma 4. The solution of problem (11) has the weak derivative ut ∈ L2 (0, T ; W0−1,2 ()). Proof. It amounts to show that for every smooth function ∈ L2 (0, T ; W01,2 ), such that (x, 0) = (x, T ) = 0, T u t dx dt M L (0,T ;W 1,2 ()) , I = 0
2
0
with a constant M independent of and u. Indeed: by virtue of identity (10) and the uniform estimate of Lemma 3 I (a ∇ u2,QT + C()f 2,QT ) ∇ 2,QT , where C() is the constant in the Poincare inequality ∀ ∈ W01,2 (),
2, C()∇ 2, .
Let us write Eq. (11) in the form ut = div G,
(19)
with G = a∇u + f ∗ ∈ L2 (QT ),
div f ∗ = f,
f ∗ |T = 0.
Eq. (19) is fulfilled in the sense of distributions. If ∇u ∈ L2 (QT ), which corresponds to the case + 0 (see Lemma 3), then according to [21, Chapter III, Lemma 4.1] the function u(x, t) satisfies the estimates
u(x + ei h, t) − u(x, t)2 h C|h|2 , i = 1, . . . , n, 2,QT (20) u(x, t + h) − u(x, t)2 h C|h|, 2,QT
where QhT = {(x, t) ∈ QT : (x + ei h, t), (x, t + h) ∈ QT , i = 1, . . . , n}, h is a scalar parameter, ei are the unit vectors of the axes xi .
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523
Lemma 5. Let u(x, t) be a solution of problem (11) and + 0. The function
u(x,t)
z(x, t) = z[u(x, t)] ≡
+
|s| ds
0
satisfies the estimates
z(x + ei h, t) − z(x, t)2 z(x, t + h) − z(x, t)2
2,QhT
2,QhT
C|h|2 ,
i = 1, . . . , n, (21)
C|h|.
+
Proof. Since ∇ z = |u| ∇ u, by virtue of (18) there holds the inclusion ∇ z ∈ L2 (QT ). The first of estimates (21) immediately follows then from the inequality |z(x + ei h, t) − z(x, t)|2 dx C, i = 1, . . . , n. |h|2 QhT To derive the second estimate we introduce the functions:
u = u(x, t + h) − u(x, t),
z = z(x, t + h) − z(x, t).
Then (z, u) = z,
z =
u1 u2
t+h t
+
|s| ds u,
∇ G(·, ) d
= − ∇(z),
t+h t
G(·, ) d
∇(z)L2 (Qh ) C.
,
(22)
(23)
T
Integrating equality (22) with respect to t ∈ (0, T − h), applying Hölder’s inequality and (23), we arrive at the inequality 2 z22,Qh T
T −h
0
C
T −h
0
t T −h
C |h| 0
T −h
|∇(z)|2 dx dt 0 t+h
t
2
|G(x, )| d
t+h
t+h
t
G(x, ) d
dx dt
dx dt
|G(x, )| d dx dt 2
C |h|.
2.3. Passage to the limit Let {u } be the sequence of solutions of problem (7). If + 0, the sequence {u } is compact by virtue of (20). If + 0, then according to (21) the sequence z[u (x, t)] is compact in Lq (QT ) with some 1 < q < ∞ [26]. The sequence {u } is also compact. Moreover, both sequences are bounded and converge a.e. in QT . The sequence {u } converges also a.e. in QT for −1 < − (x, t) + < ∞. Since u ∞,QT are uniformly bounded, we can
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S.N. Antontsev, S.I. Shmarev / Nonlinear Analysis 60 (2005) 515 – 545
extract a subsequence of the sequence {u } (which is assumed to coincide with the whole of the sequence) such that
u → u a.e. in QT , u in Lp (Q) with any 1 < p < ∞, → u strongly u dx → u dx for a.e. t ∈ (0, T ).
(24)
Consider the function g[u] = u( 2 + u2 )/2 ,
u2 + 1 ( 2 + u2 )/2 ∇ u 2 + u 2 1 + ∇ u ( 2 + u2 )/2 ln ( 2 + u2 ). 2
∇g =
If (x, t) + 1 > 0, then ∇ g ∈ L2 (QT ) because 2 2 u 1 2 2 2 2 (+1)/2 ) ln ( + u ) ∇ |∇ g|2 = ( + 1) 2 a ∇ u + + u ( 2 +u 2
C1 |a ∇ u|2 + C2 |∇ |2 , with constants Ci depending on ± and sup |u|. Then there exist functions u, g ∗ such that
g ≡ g[u ] = u ( 2 + u2 )/2 → u|u| a.e. in QT and strongly in Lq (QT ) with 1 < q < ∞, ∇g → ∇g ∗ weakly in L2 (QT ).
Let us identify the limit ∇ g ∗ . Notice that (1 + )u2 + 2 2 ( + u2 )/2 ∇ u 2 + u 2 1 = ∇ g − ∇ u ( 2 + u2 )/2 ln ( 2 + u2 ). 2
(25)
For every test-function lim ∇ g dx dt = →0 QT
∇ g ∗ dx dt QT = − lim g ∇ dx dt →0 QT = − u |u| ∇ dx dt, QT
whence ∇ g ∗ = ∇ (u|u| ) = (1 + )|u| ∇ u + 21 u|u| (ln u2 ) ∇
a.e. in QT .
(26)
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525
Given a test-function from the conditions of Definition 1, we can pass to the limit when → 0 in each term of the integral relation 5 i=1
( ) Ii
≡
−u t + ( 2 + u2 )(x,t)/2 ∇u ∇ − f dx dt
QT
+ By virtue of (24) ( ) I1 → −
QT
t=T u dx = 0. t=0
( ) I4
u t dx dt,
( ) + I5
→
t=T u dx t=0
when → 0.
Next, according to (25) and (26) 2 + u2 ( ) I2 = 2 2 QT (1 + )u +
1 2 2 /2 2 2 × ∇ g − [u ( + u ) ] ln ( + u ) ∇ ∇ dx dt 2
1 1 2 → ∇ (u|u| ) − u|u| (ln u ) ∇ ∇ dx dt +1 2 QT = |u| ∇ u ∇ dx dt when → 0. QT
Gathering these formulas we conclude that the limit function is a weak solution of problem (1)–(2). 3. Uniqueness of weak solution Theorem 2. Let the conditions of Theorem 1 be fulfilled and the function (x, t) be such that
(x, t) − > 0 in QT ,
sup |∇ | ∈ L2 (0, T ).
x∈
Then the weak solution of problem (1)–(2) is unique. Proof. In the proof we follow ideas of the papers [3,4,6–9]. Let us assume that there are two different solutions problem (1)–(2) u1 and u2 . Consider the function u = u1 − u2 . For every t ∈ (0, T ) the function u satisfies the integral identity t (−ut + (|u1 |(x,t) ∇u1 − |u2 |(x,t) ∇u2 )∇ ) dx dt 0 = − u(x, T )(x, T ) dx. (27)
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Consider the function W (x, t) ≡ W [u(x, t)] =
1 u |u|(x,t) , 1 + (x, t)
which possesses the following properties: ∇ 1+ ≡ |u| ∇ u + F (u, x, t),
∇ W (x, t) = |u| ∇ u + u |u|
ln u2 1 − 2 1+
|u1 | ∇ u1 − |u2 | ∇ u2 = (W [u1 ] − W [u2 ]) − (F (u1 , x, t) − F (u2 , x, t)). Then for every test-function :
(|u1 | ∇ u1 − |u2 | ∇ u2 ) ∇ dx dt = (∇ (W [u1 ] − W [u2 ]) ∇ + (F (u2 , x, t) − F (u1 , x, t)) ∇ ) dx dt.
QT
QT
Integrating by parts in the first term on the right-hand side, we arrive at the following representation:
(|u1 | ∇ u1 − |u2 | ∇ u2 ) ∇ dx dt = (u1 − u2 ) [−A + B ∇ ] dx dt,
QT
QT
where the vector-valued function B has the form B = A ∇ ln (1 + ) + D ∇ and 0A =
W [u1 ] − W [u2 ] 1 u1 |u1 | − u2 |u2 | = , u1 − u 2 1+ u1 − u 2
F (u1 , x, t) − F (u2 , x, t) u1 − u 2 ∇ u1 |u1 |(x,t) ln u21 − u2 |u2 |(x,t) ln u22 = − . 2(1 + ) u1 − u 2
D=
S.N. Antontsev, S.I. Shmarev / Nonlinear Analysis 60 (2005) 515 – 545
527
It is easy to verify that 0 A C,
|D| C |∇ |,
D2 |∇ |2 (u1 |u1 |(x,t) ln u21 − u2 |u2 |(x,t) ln u22 )2 C|∇ |2 , = 4(1 + ) (u1 − u2 ) (u1 |u1 | − u2 |u2 | ) A D2 C, A
B2 C|∇ |2 , A
with a constant C depending on − and max |ui |. Hence, identity (27) takes on the form u(x, T )(x, T ) dx. [−t − A + B ∇ ] u dx dt = −
QT
(28)
Let (x, t) = (x, T − t), where (x, t) is the solution of the following parabolic problem:
t − (A + ) + B ∇ = h in QT , (29)
(x, 0) = 0, = 0 on T . In these conditions > 0 is an arbitrary small parameter and h is an arbitrary function. If follows then that for every h ∈ L2 (QT ) and > 0 problem (29) has a unique continuous weak solution such that t , Dij2 ∈ L2 (QT ). Instead of identity (28) we now have: u [h + ] dx dt = 0. (30) QT
We proceed to derive a priori estimates on the derivatives of . Let h ∈ C(0, T ; C01 ()). Multiplying Eq. (29) by and integrating over we obtain the equality 1 d (31) |∇ |2 dx + (A + )| |2 dx = I1 + I2 , 2 dt where
I1 =
B ∇ dx,
I2 = −
h dx =
∇h ∇ dx.
By Hölder’s inequality 1 |B|2 1 2 |I1 | (A + )| | dx + |∇ |2 dx, 2 2 (A + ) 1 1 |I2 | |∇ |2 dx + |∇h|2 dx. 2 2 Gathering (31)–(34) we have d |∇ |2 dx + (A + )| |2 dx dt
|B|2 |∇ h|2 dx + 1 |∇ |2 dx + (A + ) ≡ J.
(32)
(33) (34)
(35)
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S.N. Antontsev, S.I. Shmarev / Nonlinear Analysis 60 (2005) 515 – 545
By the above-established properties of A and B |B|2 |B|2 C sup(|∇ |)2 , (A + ) A
(36)
with a constant C not depending on . The right-hand side of (35) is estimated as follows: J (t)
|∇ | dx + 2
|∇ h|2 dx,
where (t) = C (1 + sup |∇ |2 ) with the same constant C. Plugging this estimate into (35) and applying Gronwall’s Lemma, we have |∇ |2 dx +
t 0
(A + )| |2 dx dt C
t 0
|∇h|2 dx dt.
(37)
Gathering (30) with (37) and applying Hölder’s inequality we have that
T
0
uh dx dt =
T
T
T
0
√ √ T meas
0
1/2
0
√ √ T meas
dx dt
| |2 dx dt 1/2
( + A)| |2 dx dt
→0
when → 0,
whence 0
T
uh dx dt = 0.
Since the last relation is true for any sufficiently smooth function h, it follows that u ≡ 0. Remark 1. We leave a gap between the conditions sufficient for the existence and uniqueness of weak solutions for problem (1)–(2). The existence is proved under the assumption −1 < (x, t) < ∞, while the uniqueness theorem is true if 0 < − (x, t) < ∞. It is known that the weak solution of the Cauchy problem for Eq. (1) with the constant exponent of nonlinearity (x, t) ≡ ∈ (−1, 0) need not be unique [23] (conditions of non-uniqueness of solutions in the limit case = −1 are given in [16]).
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529
4. Finite speed of propagation 4.1. Notation and assumptions Throughout the rest of the paper we use the following notation: given a point (x0 , t0 ) ∈ QT ≡ × (0, t0 ] B ≡ B (x0 ) = {x ∈ RN : |x − x0 | < } ⊂ , S ≡ S (x0 ) = *B (x0 ). We introduce the energy functions: b(, t) = ||u(·, t)||22,B ,
E(, t) =
t 0
E() = sup
0 t t0
b(, t),
u(x, ) |∇u|2 dx d , t E(, t) = 00 B u(x, ) |∇u|2 dx d ,
B
0 t t0
E + (, t) = t +
b() = sup
0
E () = sup
+
u |∇u|2 dx d , t + E + (, t) = 00 B u |∇u|2 dx dt ≡ E + (, t0 ).
B
0 t t0
We will consider the bounded weak solutions, which is why without loss of generality we always assume that |u| 1.
(38)
Since E(, t) is monotone nondecreasing in and t, there exist the weak derivatives t E (, t) = 0 S u(x, ) |∇u|2 dS d , Et (, t) = B u(x,t) |∇u|2 dx, Et (, t) = S u(x,t) |∇u|2 dS, E () = E (t , t0 ), + + E+ (, t) = 0 S u |∇u|2 dS dt, E () = E+ (, t0 ), + + + Et (, t) = B u |∇u|2 dx, E+t (, t) = S |u| |∇u|2 dS. It will be assumed that there is a 0 > 0 such that B0 (x0 ) ⊂ and b(, 0) = B (x0 ) u2 (x, 0) dx = 0, f (x, t) ≡ 0 in B0 (x0 ) × [0, T ]. 0
All the further considerations of this section are local and restricted to the cylinder G = B (x0 ) × {t ∈ [0, ]}
with = const 0 .
4.2. Finite speed of propagation of disturbances from the data Let us introduce the notion of local weak solution of Eq. (1).
(39)
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S.N. Antontsev, S.I. Shmarev / Nonlinear Analysis 60 (2005) 515 – 545
Definition 2. A locally integrable in G bounded function u(x, t) is said to be local weak solution of Eq. (1) in G if: (i) u ∈ L∞ (0, ; L∞ (B (x0 ))), |u|(x,t)/2 ∇u ∈ L2 (0, ; L2 (B (x0 ))), (ii) for any test-function (x, t) satisfying the conditions:
∈ L2 (0, ; W01,2 (B (x0 ))) ∩ L∞ (0, ; L∞ (B (x0 ))), t ∈ L2 (0, ; L2 (B(x0 )), the following integral identity holds: G
(−ut + |u|
(x,t)
∇u∇ − f ) dx dt = −
B (x0 )
u dx .
(40)
0
Obviously, every weak solution of problem (1)–(2) is a local weak solution of Eq. (1) in the sense of Definition 2. Theorem 3 (Finite speed of propagation). Let condition (39) be fulfilled,
(x0 , 0) > 0 and (x, t) ∈ C 0 (QT ). Then there exists 0 > 0 such that then every local weak solution of Eq. (1) in G0 satisfying (38) and the condition E0 = b(0 ) + E(0 ) < ∞ possesses the property of finite speed of propagation: u(x, t) = 0
for (x, t) ∈ B(t) (x0 ) × [0, 0 ],
where (t) given by the formula
(t) = 0 − Ct E0 ,
(41)
with positive constants , , , and C = C(b(0 ), 0 , ± , 0 , N ). Remark 2. Since (t) is a monotone decreasing function with (0) = 0 > 0, the set B(t) (x0 ) is nonempty for small t. The proof follows the ideas documented in the book [5] which is devoted to the study of localization properties of solutions to nonlinear equations with constant exponents of nonlinearity. We will use the material of this book as the analytical background for the further considerations, paying a special attention to the necessary changes in the arguments due to the presence of variable exponent of nonlinearity. The proof splits into three steps: (1) justification of the integration-by-parts formula for local weak solutions of Eq. (1); (2) derivation of a nonlinear ordinary differential inequality for the energy functions associated with the local weak solution; and (3) analysis of the ordinary differential inequalities.
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531
4.2.1. Formula of integration by parts. The energy relation Lemma 6. Let G = B (x0 ) × {t ∈ (0, ]} ⊂ QT . Under the conditions of Theorem 1, every local weak solution of Eq. (1) in G satisfies the energy relation =t t t 1 b(, ) + E(, t) + f u dx d = − u |u|(x,) |∇ u|2 dx d 2 0 B 0 S =0 (42) for any ∈ (0, ] and t ∈ (0, ]. Proof. Let u be a local weak solution of Eq. (1) in G ⊂ QT with some > 0. The fact that ∇ u need not belong to L2 (G ) prevents us from taking it as the test-function in the integral identity (40), which would immediately lead us to the desired energy relation. Let us denote u+ = max{u, 0} 0,
u− = max{−u, 0} 0,
u = u+ − u− .
Fix an arbitrary > 0, introduce the functions: ± u if u± > , = u± if u± − and then set u = u+ − u . For every > 0 the functions u possess the properties [21, Chapter 2, Section 4] ∇u if |u| > , ut if |u| > , u t = (43) ∇u = 0 if |u| < , 0 if |u| < .
Moreover, by virtue of (38) ∇ u ∈ L2 (G ) because |∇u |2 dx dt = |∇ u|2 dx dt ( ) G G + + − u(x,t) |∇ u|2 dx dt C − ,
(44)
G
( )
under the notation G = G ∩ {|u| > }. Next, G
u u t dx dt =
( )
G
u ut dx dt =
G
1 u u t dx dt = 2
u dx . 2
B (x0 )
(45)
0
Given > 0 and ∈ (0, ), we define the function n (r), r = |x − x0 |, by the formula if r ∈ 0, − n1 , 1 n (r) = n( − r) if r ∈ − n1 , , n ∈ N. 0 if r ∈ [, ],
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S.N. Antontsev, S.I. Shmarev / Nonlinear Analysis 60 (2005) 515 – 545
For every > 0 the function = u n (r) can be taken for the test-function in (5), which leads to the relation 1 2 n (x)u dx + ( ) (n |u| |∇u|2 + u |u| ∇u∇ n + f n u) dx dt − 2 B G 0 =− n (x) u u dx . (46) B 0
Letting → 0 we obtain 1 2
n u dx + (n (|u| |∇u|2 + f u)) dx dt = −Jn , 0 B 2
B
(47)
0
with Jn :=
B
0
u |u| ∇u∇ n dx dt.
Passing to the spherical coordinates with the origin at the point x0 , we have Jn =
=n =n
B
0
|u|(x,t) ∇u ∇ n dx dt
0
−(1/n)<|x−x0 |<
0
− n1
→ I (, )) =
u |u| ∇ u ·
x − x0 dx dt |x − x0 |
I (r) dr dt 0
S
u |u| ∇ u · n dS dt
when n → ∞
(n denotes the exterior unit normal to S (x0 )). Passing in (47) to the limit when n → ∞ we obtain the energy integral: 1 2
u2 dx + (|u|(x,t) |∇u|2 + f u) dx dt B 0 B 0 =− u |u|(x,t) ∇u · n dS dt ≡ −I (, ).
0
S
(48)
Theorem 4. Let under the conditions of Theorem 1 + 0. Then the weak solution of problem (1)–(2) satisfies the energy relation t t t 1 2 (x,t) 2 u dx + |u| |∇u| dx dt = f u dx dt. (49) 2 0 0 0
S.N. Antontsev, S.I. Shmarev / Nonlinear Analysis 60 (2005) 515 – 545
533
− Proof. Let {u+ } and {u } be the sequences of nonnegative functions defined by the relations
u+ = max{u; } − ,
u− = max{−u; } − .
The following formulas hold: ± ∇ u if ± u > , ± ∇ u = 0 if ± u < ,
u± ;t
=
± ut 0
if ± u > , if ± u < .
Moreover, since (x, t) + 0 and |u| 1, by virtue of (15) we have 2 −(x,t) (x,t) ± 2 dx dt |∇ u | |u| |∇ u± | dx dt = |u| QT QT = |u|−(x,t) (|u|(x,t) |∇ u|2 ) dx dt |u|/2 ∇ u22,QT , Q T
1,2 where Q T = {(x, t) ∈ QT : |u| > }. Thus, u± ∈ L2 (0, T ; W0 ()). Each of the functions u± can be taken now for the test-function in the Definition 1. The corresponding integral identities read as follows: t t 1 (x,t) ± 2 + u u± dx |u| |∇u | dx dt = ± f u± dx dt. 2 Q 0 0 t
Passing to the limit when → 0, for the functions u+ = max{u; 0} and u− = max{−u; 0} we obtain the energy relations t t 2 1 (u± )2 dx + |u± |(x,t) ∇u± dx dt = ± f u± dx dt. 2 Q 0 0 t Summing these relations and taking into account the formulas: u+ + u− = |u|,
u+ − u− = u,
u2 = (u+ )2 + (u− )2 ,
|∇ u|2 = |∇ u+ |2 + |∇ u− |2 ,
we obtain (49). 4.2.2. The ordinary differential inequality The energy relation (48) is valid for any t ∈ (0, T ) and every B (x0 ) ⊂ . If assumptions (39) are fulfilled, it can be specified as follows: for ∈ (0, 0 ], t ∈ [0, T ] t 1 2 u (x, t) dx + |u|(x,t) |∇u|2 dx dt = −I (, t), (50) 2 B (x0 ) 0 B (x0 ) with |I |
t 0
S (x0 )
|u|(x,)+1 |∇u| dS d ≡ .
534
S.N. Antontsev, S.I. Shmarev / Nonlinear Analysis 60 (2005) 515 – 545 +
According to (38) |u|(x,t) |u| , and since t ∈ (0, T ) is arbitrary, we may write +
E(, t) 21 (E(, t) + E + (, t)) 21 (E() + E ()). Formula (50) can be now continued in the following way: +
b(, t) + E(, t) + E (, t) 6 .
(51)
We proceed to evaluate . Applying Hölder’s inequality and using (38) we have t 2 |u|(x,t)+2 dS dt E 0
E
t
sy (x0 )
−
S (x0 )
0
|u|2+ dS dt ≡ E 2 .
(52)
Let us make use of the trace-interpolation inequality [2]: for every fixed t ∈ (0, T ] + |u|2+ dS S
C
+
B
−
|u| |∇u| dx + 2
(2++ )/2
(b(, t))
+ )/2)(1−)
(b(, t))((2+
, (53)
with the exponents
=
N + = 2+ > 1. 2
N + + 2 ∈ (0, 1), N + + 4
In our notations, this inequality can be written as + + + |u|2+ dS C Et+ (, t) + − b(, t)(b(0 )) /2 (b(, t))((2+ )/2)(1−) S
+ C1 − Et+ (, t) + b(, t) (b(, t))((2+ )/2)(1−) ,
with the constant + C1 = C 0 + b(0 ) /2 . By Hölder’s inequality S (x0 )
|u|
2+−
dS
S (x0 )
with the exponent
= (N − 1)
+ − − 0. 2 + +
|u|
2++
(2+− )/(2++ ) dS
,
(54)
S.N. Antontsev, S.I. Shmarev / Nonlinear Analysis 60 (2005) 515 – 545
535
Gathering the above inequalities we obtain the estimate (2+− )/(2++ ) t0
2
+
S (x0 )
0
|u|2+ dS
dt
(2+− )/(2++ ) + C1 − [Et+ + b] b((2+ )/2)(1−) dt 0 t0 (2+− )/(2++ ) + dt C1 [Et+ + b] b((2+ )/2)(1−) C2 1− 0 t0 − + (2+− )(1−)/2 (Et+ + b)((2+ ))/(2+ ) dt C2 1− b t0
C3 1− t0 b
0 (2+− )(1−)/2
+
− )/(2++ ))
(E + b)((2+
+
+
≡ C3 1− t0 (E + b)2 C3 1− t0 (b + E + E )2 ,
(55)
with the constants (2+− )/(2++ )
C2 = C1
,
− )/2(2++ ))
C3 = C2 (1 + t0 )((2+
,
2 + − 2 + + + − − N + N + + 2 2 + − = − (N − 1) + 2+ , 2 + + 2 N + + 4 2 + + 2 + − 2 + − (1 − ) 2 + − , (1 − ) + + = = 4 2 2 + + 2 2 2 + + 2 + − 1 1− = > 0. 4 2 + +
= 1 − +
Let us claim that
> 0,
∈
1 ,1 2
in G0 .
(56)
Since (x, t) ∈ C 0 (QT ), (x0 , 0) > 0, and condition (56) is fulfilled for + = − = (x0 , 0), we may always choose 0 so small that (56) is fulfilled by continuity on the whole of G0 . Gathering (51), (52) and (55) and simplifying, we obtain the nonlinear ordinary differential inequality +
/2
(b() + E() + E ())1− C3 (1−)/2 t0 E . 1/2
(57) +
Keeping on the left-hand side only the term E and dropping the nonnegative terms E and b, we obtain the simplified inequality E
1−
C4 t0 1− E ,
with 0 < 1 − = 2(1 − ) < 1, C4 = C32 .
(58)
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S.N. Antontsev, S.I. Shmarev / Nonlinear Analysis 60 (2005) 515 – 545
4.2.3. Analysis of the differential inequality Integrating (58) in we have that for every ∈ (, 0 ):
0 E () E (0 ) − C5 t0− (0 − ), whence
E = sup
0 t t0
t0
E(, t) =
0
B
C5 =
, C4
(59)
u(x,t) |∇u|2 dx dt = 0,
if
(t) 0 −
t E (0 ), t ∈ [0, t0 ]. C5
(60)
By virtue of (57) we also have that b() = sup b(, t) = 0 0 t t0
and the conclusion follows: u(x, t) = 0
when x ∈ B(t) (x0 ), t ∈ [0, t0 ],
(61)
The proof of Theorem 3 is completed. 5. The waiting time effect Let us assume that in addition to condition (39) t 1/ ≡ b(, 0) + f 2 dx dt ( − 0 )+ , 0
B (x0 )
= const > 0
(62)
for ∈ [0 , 1 ] with the constant defined in the proof of Theorem 3. Theorem 5 (The waiting time property). Let the conditions of Theorem 3 be fulfilled and (62) be true. Then every local weak solution u(x, t) of Eq. (1) such that E1 =
sup
0 t 0
b(1 , 0 ) + E(1 , 0 ) < ∞
possesses the following property: there exist positive constants ∗ > 0 and t ∗ t0 T such that for every ∈ (0, ∗ ] u(x, t) = 0
in
B0 (x0 ) × [0, t ∗ ].
Proof. The energy relation (48) with = > 0 takes on the form t 1 u2 (x, t) dx + |u|(x,t) |∇u|2 dx dt = −I (, t) + J (, t), (63) 2 B (x0 ) 0 B (x0 )
S.N. Antontsev, S.I. Shmarev / Nonlinear Analysis 60 (2005) 515 – 545
with J (, t) =
1 2
B (x0 )
u2 (x, 0) dx −
t 0
B (x0 )
f u dx dt
537
(64)
and I (, t) defined in (48). Repeating the estimating performed in the proof of Theorem 3, we obtain the inequality +
+
/2
b() + E() + E () C3 −(−1)/2 t0 (E )1/2 (E + b) + |J |. The integral J is estimated in the following way: t f 2 dx dt + t0 b(), 2|J | b(, 0) + 0
(65)
t t0 .
B (x0 )
Assuming that t0 is small, from (65) we have that +
+
/2
b() + E() + E () C4 (−(−1)/2 t0 (E )1/2 (E + b) + ).
(66)
Applying Young’s inequality, we obtain +
/2
b() + E() + E () C5 ((−(−1)/2 t0 (E )1/2 )1/(1−) + ),
(67)
whence +
(b() + E() + E ())2(1−) C6 (−(−1) t0 E + 2(1−) ).
(68)
Letting 2(1− )=1− and using condition (62), we arrive at the nonhomogeneous ordinary differential inequality E
1−
(1−)/
2(1−) C(t0−1 − ( − 0 )+ 0 E +
).
Let t ∗ be chosen from the condition ∗ −1 A1− = C − 0 (t )
A + C 1− ,
with A > A0 = max{C 1− , E1 (1 − 0 )1/ }. It s easy to check that the function
E() ≡ A ( − 0 )1/ is the solution of the nonlinear ODE 1−
2(1−) E1− = C(t0−1 − ( − 0 )+ ), 0 E +
E(1 ) E1 .
The function E() is a majorant for E(), and the result follows: 1/
E() A( − 0 )+ ,
∈ [0 , 1 ].
(69)
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S.N. Antontsev, S.I. Shmarev / Nonlinear Analysis 60 (2005) 515 – 545
6. Asymptotic behaviour of solutions for large t Theorem 6. Let u(x, t) be a weak solution of problem (1)–(2) constructed in Theorem 1. Let
+ > 0.
(70)
If in Eq. (1) f (x, t) ≡ 0, then the solution u(x, t) satisfies the estimate y −1 (t)
−1
y0
−1
1 + t C0 ( − 1)y0
,
where y(t) = u(·, t)22, , y0 = y(0),
=1+
+ >1 2
and C0 is an absolute constant depending on + , , and N. If f ≡ / 0, there are positive constants K and B depending on , + and N such that if ||f (·, t)||2,
K (Bt + 1)(2−1)/2(−1)
,
then y(t)
y0 (Bt + 1)1/(−1)
.
Remark 3. If in the conditions of Theorem 1 −1 < − (x, t) + 0, we satisfy condition (70) choosing for + any positive number. Proof. Let u(x, t) be a weak solution of problem (1)–(2). For every t ∈ (0, T ) the energy relation (49) holds: t2 t2 t2 1 u2 dx + |u|(x,t) |∇u|2 dx dt = f u dx dt. (71) 2 t1 t1 t1 By the method of construction, every weak solution satisfies (|u|(x,t) |∇u|2 − f u) dx ∈ L1 [0, T ],
which allows us to differentiate (71) with respect to t: for a.e. t ∈ (0, T ) 1 d u2 (x, t) dx + |u(x, t)|(x,t) |∇u(x, t)|2 dx = − f u dx. 2 dt
(72)
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539
Recall that the weak solutions are bounded. Not loosing generality we may assume therefore that |u| 1. Applying the imbedding theorem [15, Theorem 7.10], we have: for every 2N if N 2, 1 p N−2 ∞ if N = 1, there is a constant C ≡ C(p, N, + , ) such that 2/p + p(+ +2)/2 |u| dx C |∇(u( +2)/2 )|2 dx + |u|(x,t) |∇u|2 dx. =C |u| |∇u|2 dx C
Applying Hölder’s inequality we now have that
y(t) ≡
u2 dx C
C0
|u|
+ +2)/2
(x,t)
up(
|∇u| dx 2
4/p(2++ ) dx
2/(2++ )
,
(73)
with a new constant C0 = C0 (p, + , ). Substituting this inequality into (72), we have y + C0 y (t) , with
(74)
= f u dx ,
=
+ + 2 > 1. 2
(75)
Let f ≡ 0. Integrating inequality (74), we come to the following estimate on the rate of vanishing of the L2 ()-norm of the weak solution: y −1 (t)
−1
y0
−1
1 + tC 0 ( − 1)y0
→0
when t → ∞,
where y0 = u(x, 0)22, . Let f ≡ / 0. Applying Hölder’s inequality we estimate y(t) f (·, t)2, and arrive at the following nonhomogeneous ordinary differential inequality: √ (76) y + C0 y (t) yf (·, t)2, . √ Introduce the new thought function u(t) = y(t). This function satisfies the ordinary differential inequality u + 21 C0 u2−1 21 ||f (·, t)||2, .
(77)
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S.N. Antontsev, S.I. Shmarev / Nonlinear Analysis 60 (2005) 515 – 545
Let ||f (·, t)||2,
K (Bt + 1)(2−1)/2(−1)
,
(78)
with the constants (2−1)/2
K < C0 y0
,
(2−1)/2
B = (C0 y0
−1/2
− K)( − 1)y0
> 0.
(79)
Then the function √ w(t) = y0 (Bt + 1)−1/2(−1) solves the nonlinear ODE w + 21 Cw 2−1 (t) = 0,
w(0) =
√
y0
and is a majorant for the function u: u(t) w(t). Reverting to the function y(t) we get the estimate −1/(−1) |u(x, t)|2 dx y0 (Bt + 1) . (80) y(t) =
7. Vanishing of solutions in a finite time Theorem 7. Let the conditions of Theorem 1 be fulfilled and u(x, t) be a weak solution of problem (1)–(2). Let −1 < − (x, t) + < 0. If f (x, t) ≡ 0, then u(x, t) vanishes in the finite time: u(x, t) = 0
in for t t ∗ ,
where t ∗ depends only on u0 2, , + , and . Let f ≡ / 0. There exist > 0 and t > t ∗ such that if + +
t −(1+ )/ f (·, t)2, 1 − tf +
with tf t and 0 < < , Then u(x, t) ≡ 0
for t tf .
Proof. Arguing like in the previous section, for the function u2 (x, t) dt, y(t) =
(81)
S.N. Antontsev, S.I. Shmarev / Nonlinear Analysis 60 (2005) 515 – 545
541
we derive the ordinary differential inequality dy + Cy (t) , dt
(82)
with the right-hand side = f u dx ,
the exponent
+ <1 2 and a finite constant C which is defined through the constant in the embedding theorem. Let f ≡ 0. Integrating (82) we arrive at the estimate =1+
0 y 1− (t) y 1− (0) − C(1 − )t, whence y(t) ≡ 0
for t t ∗ =
y 1− (0) <∞ C(1 − )
(83)
and, consequently, u(x, t) = 0 for t t ∗ . For f ≡ / 0 the ordinary differential inequality for y(t) becomes √ y + Cy yf (·, t)2, . √ Introducing the function u(t) = y(t) we write the last inequality in the form u + 21 Cu (t) 21 ||f (·, t)||2, ,
= 2 − 1 = + + 1 < 1,
(84)
where f satisfies condition (81). Let us consider the ordinary differential equation + +
t −(1+ )/ 1 v + C v (t) = 1− . 2 2 tf +
(85)
The solutions to this ODE are majorants for u(t). It is easy to check that the function +
t −1/ u(t) = u(0) 1 − tf + satisfies (85) provided that the parameters u(0), ε, and tf are related by the condition −
1 1 ε 1 + + u(0) + Cu−(1+ )/ (0) − 0. 1− tf 2 2
Given u(0), we satisfy this condition increasing tf and decreasing .
(86)
Corollary 5. It follows from the proof that the presence of the effect of finite time vanishing of solutions is guaranteed by fulfillment of condition (86) which connects the three
542
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parameters: —the “intensity” of the right-hand side in Eq. (1), tf —the moment when the right-hand side vanishes, and u0 2, . The assertion of Theorem 7 remains true if we fix any of the three parameters and vary the other two in order to satisfy condition (86).
8. A limit case in the finite time vanishing: + = 0 The condition + < 0 was crucial for the proof of Theorem 7. However, this condition is sufficient but not necessary for the solution to extinct in finite time. A solution of problem (1)–(2) may extinct in finite time even if + = sup (x, t) = 0. Let us consider the following problem:
ut − (u(x) ux )x = 0 in QT = (0, 1) × (0, T ), u(0, t) = u(1, t) = 0, u(x, 0) = u0 (x) 0,
(87)
where the exponent ≡ (x) satisfies the conditions
−1 < − (x) + = 0 for x ∈ [0, 1], (x0 ) = 0 with either x0 = 0 or x0 = 1, ( (x))2 20 > 0, (x) 0.
(88)
By Theorem 1, under these conditions on (x) problem (87) has a nonnegative weak solution u(x, t) for every nonnegative u0 ∈ L2 (0, 1). Theorem 8. Let conditions (88) be fulfilled, u0 ∈ L2 (0, 1), and the condition 0 u0 1 holds. Then any nonnegative weak solution u(x, t) of problem (87) vanishes in the finite time: u(x, t) ≡ 0 for x ∈ [0, 1] and t t∗ = −
1 8 . 2 0 ln y(0)
Proof. The nonnegative weak solution of problem (87) satisfies the energy relation (72) 1 d 2 dt
1
1
u2 (x, t) dx +
0
0
u(x, t)(x) u2x (x, t) dx = 0,
(89)
whence y(t) ≡
1
u2 (x, t) dx y(0) < 1.
0
Moreover, 0 u(x, t) 1 in QT . Let us introduce the new function
u1+(x)/2
(x, t) ≡ [u(x, t)] = 0
ln d 0,
(0, t) = (1, t) = 0.
(90)
S.N. Antontsev, S.I. Shmarev / Nonlinear Analysis 60 (2005) 515 – 545
543
A direct calculation gives:
2 ( )2 2+ 2 u ln u, u ux = (u1+/2 )2x − u1+/2 (u1+/2 )x ln u + 4 2 1+/2 x = 1 + )x u1+/2 ln u. (u 2 Gathering these two formulas we have that 1 ( )2 2+ 2 2 1+/2 2 u ux = (u )x + ln u − . u 4 1 + /2 x (1 + /2)2
1+
Since the function (x) is convex and 0, we also have that 1 1 x dx = − dx 0. (1 + /2)3 (1 + /2)3 0 0 Writing the energy relation in the form 1 1 ( )2 2+ 2 1 d 1 2 1+/2 2 (u u u (x, t) dx + )x + ln u dx 2 4 2 dt 0 0 (1 + /2) 1 + dx = 0. (1 + /2)3 0 and then dropping the nonnegative terms, we arrive at the inequality 1 1 ( )2 2+ 2 1 d 1 2 u (x, t) dx + ln u dx 0. u 2 2 dt 0 4 0 (1 + /2) In our assumptions on (x) and u(x, t), this inequality can continued as follows: 20 1 2 2 2 1 d 1 2 u (x, t) dx + u ln u dx 0. 2 dt 0 16 0 Let us consider now the function (s) ≡ s ln2 s. Since 2 (s) = (ln s + 1), s this function is convex on the interval s ∈ (0, 1/e). By Jensen’s inequality 1 1 1 u2 dx ≡ u2 dx ln2 u2 dx 0 0 0 1 1 (u2 ) dx ≡ u2 ln2 u2 dx
(s) = ln2 s + 2 ln s,
0
0
and (91) becomes 1 1 20 1 d 1 2 2 2 2 u (x, t) dx + u dx ln u dx 0. 2 dt 0 16 0 0
(91)
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S.N. Antontsev, S.I. Shmarev / Nonlinear Analysis 60 (2005) 515 – 545
Thus, the function y(t) = u(·, t)22,(0,1) satisfies the ordinary differential inequality
20 y ln2 y 0, 8
y +
y(0) < 1.
(92)
Any nonnegative solution of the ODE z +
20 z ln2 z = 0, 8
z(0) = y(0) < 1,
is a majorant for the function y(t). The function z(t) can be constructed explicitly: 1 . z(t) = exp 20 1 + t ln y(0) 8 This is a monotone decreasing function that vanishes at the instant t∗ = −
1 8 ∈ (0, ∞). 2 0 ln y(0)
Since z(t) y(t) 0, the assertion of theorem follows.
Remark 4. It is easy to see that unless we impose some additional assumptions on the data, the assertion of Theorem 8 is false if the function (x) ≡ 0 on an interval I =[a, b] ⊂ [0, 1]. Indeed, let us assume the contrary: (x) ∈ (−1, 0], (x) ≡ 0 for x ∈ [a, b] ⊂ [0, 1], and u(x, t) ≡ 0 for all x ∈ [0, 1], t ∈ [t∗ , T ]. We may consider u(x, t) as a nonnegative solution of the Dirichlet problem for the heat equation in the rectangle (a, b) × (0, t∗ ), so that the assumption u(x, t∗ ) ≡ 0 on [a, b] contradicts the strong maximum principle. Acknowledgements The first author was partially supported by the Project DECONT—Differential Equations of Continuum Mechanics, FCT/MCES (Portugal) at the “Centro de Matemática”, Universidade da Beira Interior, and by the Project FCT-POCTI /MAT/34471/2000. The second author was supported by the Spanish Research Project BFM2000-1324 and by the European RTN Programme HPRN-CT-2002-00274. References [1] A. Acerbi, G. Mignione, G. Seregin, Regularity results for a class of parabolic systems related to a class of non-newtonian fluids, Ann. Inst. H. Poincaré 21 (6) (2004) 25–60. [2] S.N. Antontsev, Localization of solutions of degenerate equations of continuum mechanics, Akademiya Nauk SSSR, Sibirsk. Otdel., Institut Gidrodinamiki, Novosibirsk, 1986 (in Russian). [3] S.N. Antontsev, J.I. Díaz, A.V. Domanski˘ı, Stability and stabilization of generalized solutions of degenerate problems of two-phase filtration, Dokl. Akad. Nauk 325 (1992) 1151–1155 (English Trans. in Soviet Phys. Dokl. 37(8) (1992) 411–413).
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