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Asymptotic behaviour of a quasilinear hyperbolic equation with hysteresis
Nonlinear Analysis: Real World Applications 8 (2007) 1398 – 1409 www.elsevier.com/locate/na
Asymptotic behaviour of a quasilinear hyperbolic equation with hysteresis夡 P. Kordulová∗ Mathematical Institute, Silesian University in Opava, 746 01 Opava, Czech Republic Received 31 May 2006; accepted 19 July 2006
Abstract The paper deals with the asymptotic behaviour of the solution of a quasilinear hyperbolic equation with hysteresis. A stability result for solution in L1 () is derived by the nonlinear semigroup approach. 䉷 2006 Elsevier Ltd. All rights reserved. MSC: 34C55; 47J40; 35L60 Keywords: Hysteresis operator; Asymptotic behaviour
1. Introduction We study the asymptotic stability of the equation j(u + v) ju + =0 jt jx u(0, t) = 0,
in (0, L) × [0, T ], (1)
where v = Fr [u] exhibits hysteresis of a type of a generalized play. Hysteresis is represented by functional describing adsorption and desorption on the particles of substance. We introduce here a generalized play operator as a model for adsorption–desorption [8,9]. The motivation comes from applications in chemical and geological engineering; see [7,8]. The purpose of this work is to study asymptotic behaviour of the solution. Firstly, we transform our equation with hysteresis into a system of differential inclusions containing an accretive operator. We investigate the solution based on the theory of nonlinear semigroups, which is used for solving equations with hysteresis [1,3,5,10]. We devote a large part of the paper to a proof of m- and T-accretivity of the operator provided that Fr [.] is generalized play operator, the existence and uniqueness of integral solution for (1) follows. It was proved that its unique integral solution fulfils the entropy condition. In order to get the stability result we apply the theorem of Wittbold [4,11]. 夡 The research was supported, in part, by project GD 201/03/H152 from the Czech Science Foundation and MSM4781305904 from the Czech Ministry of Education. ∗ Fax: +420 553684680. E-mail address: [email protected].
1468-1218/$ - see front matter 䉷 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.nonrwa.2006.07.007
P. Kordulová / Nonlinear Analysis: Real World Applications 8 (2007) 1398 – 1409
1399
These results can be extended to possibly discontinuous generalized Prandtl–Ishlinskii operators of play type. This includes the case of possibly discontinuous Preisach operators. The paper is organized as follows. In Section 2 the concept of hysteresis is introduced. We define the generalized play operator and describe its hysteresis behaviour; see [2,10]. This definition of the generalized play operator includes discontinuous relay operator. We give definitions of a stationary supersolution and a stationary subsolution and recall the asymptotic result of Wittbold in Section 3. Section 4 is devoted to a semigroup approach and contains important theorem about m- and T-accretivity of the operator together with its proof. Moreover, a theorem about existence and uniqueness of integral solutions [10] is recalled. In Section 5 we apply the theorem of Wittbold together with the theorem about accretivity for hyperbolic operators with hysteresis to get the asymptotic behaviour of the solution. 2. Hysteresis play operator Hysteresis is a phenomenon in which the response of a physical system to an external influence depends not only on the present magnitude of the influence but also on the previous history of the system. Hysteresis operators are characterized by two main properties—memory effect and rate independence. In this section we recall definition of the generalized play operator [2,10]. Formally, let l , r : R → P( R) be maximal monotone (possibly multivalued) functions with inf r (u) sup l (u)
∀u ∈ R.
(2)
We set R := [−∞, +∞]. The corresponding input–output behaviour is the following (see Fig. 1): when u is increasing and a given initial value v0 (x) ∈ r (u), v follows the right hysteresis boundary curve. When u starts decreasing, we will move inside the hysteresis loop and v(t) is constant till the moment when u hits the left hysteresis boundary curve and u decreases on it. If u is increasing again, we move inside the loop again, etc. Let v0 ∈ R is given, u be a continuous piecewise linear function [0, T ] → R (namely, function whose graph is a polygonal). For any t ∈ [0, T ], let t0 = 0 < t1 < · · · < tN = T be such that u is linear (or, more precisely, affine) in [ti−1 , ti ] for i = 1, 2, . . .. Then we set recursively if t = 0, min{l (u(0)), max{r (u(0)), v0 }} Fr (t) = min{l (u(t)), max{r (u(t)), v(ti−1 )}} if t ∈ (ti−1 , ti ], i = 1, 2, . . . . Fr is called generalized play (Fig. 1). The hysteresis relation is assumed to hold pointwise in space, v(x, t) = [Fr (u(x, .), v0 (x))](t) in [0, T ] a.e. in ,
v
γl
γr
Fig. 1. Generalized play.
u
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P. Kordulová / Nonlinear Analysis: Real World Applications 8 (2007) 1398 – 1409
where Fr is a hysteresis operator, v0 (x) a given initial value, let be an open bounded subset of RN (N 1) of Lipschitz class. 3. The sub/supersolution method In this chapter we recall results about the asymptotic behaviour of the equation ut + Au 0,
u(0) = u0 ,
(3)
where let A be an m- and T-accretive operator in L1 (). Definition 3.1. A stationary supersolution of (3) is defined to be a function v ∈ L1 () satisfying u0 v,
a.e. on ,
(4)
and (I + A)−1 v v,
a.e. on , ∀ > 0.
(5)
A stationary subsolution of (3) is defined in the same way with reversed inequalities. Remark 3.1. Note that if v ∈ D(A) and if A is single valued, then (4) and (5) are equivalent to v u0 , a.e. on , Av 0, a.e. on .
(6) (7)
Also note that if A is an accretive operator, (6) and (7) imply (4) and (5). Theorem 3.1 (Wittbold [11]). Let A be an m- and T-accretive operator in L1 (), i.e., R(I + A) = L1 (),
∀ 0,
(8)
and ˜ + 1 , (u − u) ˜ + 1 , (JA (u) − JA (u))
∀0, u, u˜ ∈ L1 (),
(9)
where JA (u) = (I + A)−1 (u),
and R denotes the range.
(10)
Suppose that A−1 0 = {0} and let u0 ∈ D(A). Then the following holds: if there exist a stationary subsolution and a stationary supersolution of ut + Au 0, u(0) = u0 , in L1 (), then the solution u of (3) is stable, i.e., u(x, t)1 → 0 as t → ∞. The existence of sub/supersolutions alone implies stability in L1 (). In particular, we do not need the full strength of accretivity of operators to apply this method, but only that resolvents are order preserving. 4. Semigroup approach Here we study the accretivity properties of the generalized play operator for equation (1) in × [0, T ], where is represented by (0, L). We denote by R21 the Banach space of vectors U := (u, v) ∈ R2 , endowed with the norm (u, v)R2 := |u| + |v| ∀(u, v) ∈ R21 . 1
P. Kordulová / Nonlinear Analysis: Real World Applications 8 (2007) 1398 – 1409
L1 (, R21 ) is a Banach space endowed with the norm U L1 (,R2 ) := (|u(x)| + |v(x)|) dx 1
∀ U := (u, v) ∈ L1 (, R21 ).
1401
(11)
The space H is defined as H := {u ∈ H 1 (), u(0) = 0 in the sense of traces}. In the space L2 (, R21 ) the norm is denoted by .2 and the scalar product by ., . . The norm in H is denoted by .H Eq. (1) is equivalent to the Cauchy problem: ⎧ ⎨ jU + A(U ) + R(U ) 0, in × [0, T ] jt (12) ⎩ U (0) = U0 , where D(A) := {U := (u, v) ∈ R2 : inf r (u) v sup l (u)}, A(U ) := {(, −) ∈ R2 : ∈ (U ) ∩ R} B(u) :=
∀U ∈ D(A),
(13)
ju , jx
R(U ) := (B(u), 0), D(R) := {U ∈ L1 (, R21 ) : Bu ∈ L1 ()}, Q(U ) := A(U ) + R(U ), D(Q) := {U := (u, v) ∈ L1 (, R21 ) : U ∈ D(A) a.e. in , u ∈ W 1,1 (), u(0) = 0},
(14)
and by setting U := (u, v), U0 := (u0 , v0 ), ⎧ {+∞} if v < inf r (u), ⎪ ⎪ ⎪ + ⎪ ⎪ if v ∈ r (u)\l (u), R ⎪ ⎪ ⎪ ⎨ {0} if sup r (u) < v < inf l (u), (u, v) = − ⎪ ⎪ if v ∈ l (u)\r (u), R ⎪ ⎪ ⎪ ⎪ ⎪ {−∞} if v > sup l (u), ⎪ ⎩ R if v ∈ l (u) ∩ r (u).
(15)
+ − Here R := [−∞, +∞], R := [0, +∞], R := [−∞, 0].
R) is such Proposition 4.1 (Accretivity. General case). Assume that the (possibly multivalued) function
: R2 → P( that D(
) := {(u, v) ∈ R2 :
(u, v) ∩ R = ∅} = ∅, and
Set
) ∀ (ui , vi ) ∈ D(
if u1 < u2 and v1 > v2
∀ i ∈
(ui , vi )(i = 1, 2), then 1 2 .
(u, v) ∩ R = ∅}, D(Aˆ ) := {U := (u, v) ∈ R2 :
ˆ ) := {(, −) ∈ R2 : ∈
ˆ A(U (u, v) ∩ R} ∀U ∈ D(A).
(16)
(17)
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P. Kordulová / Nonlinear Analysis: Real World Applications 8 (2007) 1398 – 1409
Then Aˆ is T-accretive in R21 . Moreover, if (u, v)} ∃ aˆ > 0 : ∀z ∈ R, Gz : u → {v ∈ R : v − z ∈ aˆ
is a maximal monotone (possibly multivalued) function in R2 ,
(18)
then Aˆ is m-accretive in R21 . Proposition 4.1 is proved in Visintin [10]. Theorem 4.1 (Accretivity. Rate independent case). Assume that l and r fulfil (2) and that D(A) = ∅. Then the operator A is T- and m-accretive in R21 . Proof. Direct application of Proposition 4.1, as condition (18) is fulfilled for any aˆ > 0. In fact, for any (u, z) ∈ R2 and any aˆ > 0, Gz defined in (18) as follows: ⎧ if z inf r (u), Gz (u) = r (u) ⎪ ⎪ ⎪ ⎨ Gz (u) = [z, sup r (u)] if inf r (u) < z < sup r (u), (19) if sup r (u) z inf l (u), Gz (u) = {z} ⎪ ⎪ ⎪ Gz (u) = [inf l (u), z] if inf l (u) < z < sup l (u), ⎩ Gz (u) = l (u) if sup l (u) z; is a maximal monotone (possibly multivalued) function.
Theorem 4.2. Assume that l (u), r (u) fulfil (2) and are affinely bounded, that is, there exist constants C1 , C2 > 0, such that ∀w ∈ R, ∀z ∈ h (w) |z| C1 |w| + C2 ,
(h = l, r).
(20)
Let A and R be defined as previously. Then the operator A + R is m- and T-accretive in L1 (, R21 ). Proof. Visintin [10] just outlined the argument, which is similar to that used for the linear second order elliptic operator in divergence form. The main difference is in the first part of the proof. Here the whole proof is made in detail. We split it into several steps. In the first three steps we study our operator in L2 (, R21 ). In the first step we use Yosida approximation of curves r and l and for any n ∈ N . We prove m-accretivity of the operator Q. In the second one we show T-accretivity of the operator, by using the Heaviside graph, for any n ∈ N . In the next step we take the limit in n and get the same properties also for the limit operator. Finally it is studied in L1 (, R21 ). (1) Let us consider the Yosida approximation of r and l ,
−1
−1 1 1 , ln := n I − I + l ∀n ∈ N , rn := n I − I + r n n and define the corresponding An and n as in (13), (15). Let us fix any n ∈ N. We claim that, if the positive constant a is sufficiently small, then for any F := (f1 , f2 ) ∈ L2 (, R21 ) there exist un ∈ H01 (), vn , n ∈ L2 () such that Bun ∈ L2 () and un + an + aBun = f1 a.e. in , (21) a.e. in , vn − an = f2 n ∈ (un , vn ) a.e. in . To prove this statement, let us define Gn := Gf2 ,n as in (19); this is a single-valued function R → R. By eliminating n and vn, the latter system becomes un + Gn (un ) + aBun = f1 + f2
a.e. in .
(22)
P. Kordulová / Nonlinear Analysis: Real World Applications 8 (2007) 1398 – 1409
The functional a : u¯ →
1403
(u¯ + aBu) ¯ u¯ dx
is not coercive for any a > 0. Therefore, we approximate R by means of R − ε, where ε is a positive parameter and D(R − ε) = D(R) ∩ H. The corresponding functional aε is coercive and a is lower semicontinuous. This yields an equation of the form un + Gn (un ) + a(B − ε)un = f1 + f2
a.e. in .
(23)
In fact, we have u¯ 2 + a [u¯ + a(B − ε)u] ¯ u¯ dx = = u¯ 2 dx +
j2 u¯ ju¯ u¯ − aε 2 u¯ dx jx jx ju¯ 2 j2 u¯ 1 a aε 2 u¯ dx dx − jx 2 jx 2 1 ju¯ = u¯ 2 + aε dx + a[u¯ 2 (x)]L 0 jx 2 2 ju¯ L ju¯ 2 u¯ + aε dx − aε u¯ = jx 0 jx 1 ju¯ + a u¯ 2 (L) − aε (L)u(L). ¯ 2 jx
The first addendum is positive, the last two addendums give a constant c. According to the definition of coercivity we have if u ¯ H → ∞: 2 2 + ju¯ + c u ¯ jx 2 u¯ + aBu, ¯ u
¯ 2 = lim lim u ¯ H →∞ u ¯ H →∞ u ¯ H u ¯ H u ¯ 2H = lim = lim u ¯ H = ∞. u ¯ H →∞ u ¯ H →∞ ¯ H u Hence by standard results there exists a unique solution un ∈ H of (23). We take the limit as ε → 0 in (23) and from it we get the existence and uniqueness of a solution of (22). Then vn = Gn (un ), and n is determined by the second equation of (21). (2) We claim that for any n ∈ N and for any F1 , F2 ∈ L2 (, R21 ), setting Ui := (I + aAn + aR)−1 (Fi ) (i = 1, 2), (U1 − U2 )+ L1 (,R2 ) (F1 − F2 )+ L1 (,R2 ) . 1
1
(24)
Then the analogous inequality holds for the negative parts, whence (U1 − U2 )L1 (,R2 ) (F1 − F2 )L1 (,R2 ) . 1
1
(25)
In order to prove (24), first let us take any > 0, set ⎧ ⎪ ⎨ 0u¯ if u¯ < 0, ¯ = H (u) if 0 u¯ , ∀u¯ ∈ R, ⎪ ⎩ 1 if u¯ > , and denote by H the Heaviside graph. In this argument we omit the (fixed) index n. For i = 1, 2, let us denote by (ui , vi , i ) the solution of Eq. (21) corresponding to Fi and set U˜ = (u, ˜ v) ˜ := U1 − U2 , ˜ := 1 − 2 .
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P. Kordulová / Nonlinear Analysis: Real World Applications 8 (2007) 1398 – 1409
For any measurable function h1 such that h1 ∈ H (v) ˜ a.e. in , by (21) we have ˜ + ] dx (F1 − F2 )+ L1 (,R2 ) = [(u˜ + a ˜ + aBu) ˜ + + (v˜ − a ) 1 ˜ 1 ] dx [(u˜ + a ˜ + aBu)H ˜ (u) ˜ + (v˜ − a )h ˜ (u) ˜ − h1 ]} dx {uH ˜ (u) ˜ + vh ˜ 1 + a [H +a ˜ dx. BuH ˜ (u)
Note that
H (u) ˜ → h0 = Hence
0 1
if u˜ 0 if u˜ > 0
(26)
a.e. in .
[uH ˜ (u) ˜ + vh ˜ 1 ] dx → (u˜ + + v˜ + ) dx = U˜ + L1 (,R2 ) , 1 ˜ 0 − h1 ) dx. ˜ (u) (h [H ˜ − h1 ] dx →
Since rn , ln are single-valued, then ∀u, v1 , v2 ∈ R, if v1 < v2 , then ∀1 ∈
(u, v1 ), ∀2 ∈
(u, v2 ), 1 2 holds. Then we have ∃h1 ∈ H (v) ˜ : ∀h0 ∈ H (u), ˜
˜ 0 − h1 ) 0 a.e. in . (h
Hence, for this choice of h1 , the limit as → 0 of the first integral of the last member of (26) is not smaller than U˜ + L1 (,R2 ) . 1 u¯ Now let us consider the last integral of (26). To deal with the first order term, let us set (u) ¯ := 0 H () d for any u¯ ∈ R; thus ¯ 0 (u)(1/)
d = /2.
0
So we have ju˜ ˜ j ju H (u) dx = − (u) ˜ dx = − uH ˜ ˜ dx 0. jx jx jx Hence we conclude that lim inf →0 BuH ˜ (u) ˜ dx 0. Therefore, (24) holds. (3) Let us now take the limit in n. Multiplying (23) by un and integrating in we get 2 ju¯ 2 un + Gn (un )un + aε + c dx f1 + f2 L2 () un H jx un 2H − c3 un H − c2 f1 + f2 L2 () un H c2 un H − c3 − f1 + f2 L2 () . un H Now we have two cases: (a) un H is unbounded. It is impossible due to assumption of f1 + f2 , (b) un H is bounded. So that un H constant (independently of n).
(27)
P. Kordulová / Nonlinear Analysis: Real World Applications 8 (2007) 1398 – 1409
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So it is easy to see that un H ,
vn L2 () constant (independently of n).
Hence there exist u, v ∈ L2 () such that, possibly taking n → ∞ along a subsequence, un → u weakly in H,
(28)
vn → v
(29)
weakly in L2 ().
u¯ Setting n (u) ¯ := 0 Gn (y) dy for any u¯ ∈ R, (22) is equivalent to the following variational inequality: [(un + aBun − f1 − f2 )(un − v) + n (un ) − n (v)] dx 0
∀v ∈ L2 ().
(30)
Let us denote by a lower semicontinuous convex function such that j = Gf2 (j denoting the subdifferential of
). Because of the lower semicontinuity of , u solves the limit inequality, obtained from (30) by replacing n by . We employ here the compactness of the inclusion H ⊂ L2 (). We take the limit as ε → 0 in (23). Hence we can pass to the limit in (22), getting v := −u − aBu + f1 + f2 ∈ Gf2 (u)
a.e. in .
Therefore, setting := (1/a)(v − f2 ) ∈ L2 (), we get
u + a + aBu = f1 v − a = f2 ∈ (u, v)
a.e. in , a.e. in , a.e. in .
(31)
Note that, by comparison of the terms of (31)1 , Bu ∈ L2 (), and this equation holds a.e. in . Also for this limit problem the solution is unique. When we take the limit in (24) and (25), we get the T-accretivity also for the limit operator A + R. (4) Finally, we have to study our operator in L1 (, R21 ). Let us fix any F ∈ L1 (, R21 ), any sequence {Fm ∈ 2 L (, R21 )} such that Fm → F
strongly in L1 (, R21 ),
and set Um := (I + aA + aR)−1 (Fm )(∈ L2 (, R21 )). By (25), there exists U ∈ L1 (, R21 ) such that Um := (um , vm ) → U := (u, v)
strongly in L1 (, R21 );
this yields vm − f2m v − f2 → =: strongly in L1 (), a a aBum = f1m − um − am → f1 − u − a = aBu strongly in L1 (). m =
Therefore, (u, v, ) solves the corresponding problem (31). Then we can pass to the limit in the inequalities (24) and (25), provided that we show that the solution of (31) is unique. To this aim, let us fix any f1 , f2 ∈ L1 (), consider two solutions (ui , vi , i ) (i = 1, 2), take the difference between the system (31) written for i = 1, 2, and set u˜ := u1 − u2 , v˜ := v1 − v2 . By eliminating 1 and 2 , we get u˜ + v˜ + aBu˜ = 0 a.e. in , (32) vi ∈ Gf2 (ui )(i = 1, 2) a.e. in .
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P. Kordulová / Nonlinear Analysis: Real World Applications 8 (2007) 1398 – 1409
Now let us multiply the first equation by any measurable function s such that s ∈ sign(u˜ + v) ˜ a.e. in . Since Gf2 is nondecreasing, we have s ∈ sign(u); ˜ hence, setting sj (y) := arctan(jy) for any j ∈ N and any y ∈ R, we get ˜ dx = −a lim (Bu)s ˜ j (u) ˜ dx. |u˜ + v| ˜ dx = −a (Bu)s
j →∞
By a calculation similar to (27), we get that the latter limit is nonnegative. This yields the desired uniqueness property. As we saw, the operator occurring in the Cauchy problem is m- and T-accretive. Here we apply some classical results of the theory of nonlinear semigroups (Theorem 4.3). Theorem 4.3. Let B be a Banach space and Z : D(Z) ⊂ B → B be an m-accretive operator. Then (i) If u0 ∈ D(Z), then Cauchy problem has one and only one integral solution. (ii) The integral solution depends Lipschitz continuously on the data u0 . (iii) If u0 ∈ D(Z), then the integral solution of Cauchy problem is Lipschitz continuous in [0, T ]. Theorem 4.4. Let be an open subset of R of Lipschitz class. Let L1 (, R21 ) be endowed with the norm (11). Define the operator R as in (14). Let A be as above, and assume that (20) holds. Take any U0 := (u0 , v0 ) ∈ L1 (, R21 ) such that U0 ∈ D() a.e. in , and set Q := A + R. Then the Cauchy problem (12) has one and only one integral solution U : [0, T ] → L1 (, R21 ), which depends continuously on data u0 , v0 . Moreover, if Ru0 ∈ L1 (), then U is Lipschitz continuous. 5. Main result In this section we prove two new lemmas and apply the above results to get the asymptotic behaviour of the solution, the main result of the paper. Theorem 5.1. Suppose all conditions of Theorem 4.2 are satisfied, i.e., the operator A + R is m- and T-accretive in L1 (, R21 ). Suppose also that u0 ∈ D(Q). Then there exists v∞ (x) dependent on x only, such that for the solution u U= v of jU + Q(U ) = 0, jt u0 U (0) = , v0 the following holds: .1 − lim u(x, t) = 0, t→∞
.1 − lim v(x, t) = v∞ (x). t→∞
Before proving this theorem we will prove two lemmas which will be used in the proof of the theorem. Lemma 5.1. Let Q be m- and T-accretive operator in L1 (, R21 ). Then the following holds:
0 Q−1 0 = , such that inf r (0)v(x) sup l (0) . v(x)
(33) (34)
P. Kordulová / Nonlinear Analysis: Real World Applications 8 (2007) 1398 – 1409
1407
Proof.
ju + jx u = ; ∈ (U ) ∩ R , v − u so that v ∈ Q−1 0 if and only if the following is satisfied: Q
ju = 0, jx − = 0,
+
which implies = 0, which is satisfied by any v(x), if inf r (u(x))v(x) sup l (u(x)). We then must have ju = 0 on , jx u(0, t) = 0. Therefore, u = 0 is the unique solution of the previous (preceding) equation, and the assertion follows.
Lemma 5.2. If u0 ∈ L∞ (), there exist a stationary supersolution and a stationary subsolution of the equation jU/jt + Q(U ) 0. Moreover, those can be chosen so that they belong to D(Q). Proof. We show only the existence of a stationary supersolution. The proof of existence of a subsolution is analogous. We can look for solutions of +
ju 0, jx
a.e. on ,
−0,
a.e. on ,
u u0 ,
a.e. on ,
v v0 ,
a.e. on .
(35)
We can choose = 0. Then from the first equation and the third one we require ju 0, jx
a.e. on ,
(36)
u u0 ,
a.e. on .
(37)
On the assumption that u0 ∈ L∞ (), u can be easily found, in fact u = u0 L∞ satisfies ju = 0, jx
a.e. on ,
u u0 , a.e. on . u Then v will be the supersolution which can be found if we choose v such that inf r (u) < v < sup l (u). The last assertion in the lemma is obvious. Proof of the main Theorem. Suppose first that all conditions of Theorem 5.1 are satisfied, i.e., the operator Q is mand T-accretive in L1 (, R21 ), R(I + Q) = L1 (, R21 ),
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P. Kordulová / Nonlinear Analysis: Real World Applications 8 (2007) 1398 – 1409
and (JQ (U ) − JQ (U˜ ))+ 1 (U − U˜ )+ 1 ,
(38)
where JQ (U ) = (I + Q)−1 (U ) and
U 1 =
(|u(x)| + |v(x)|) dx,
denotes the norm in L1 (, R21 ). Furthermore, we suppose that u0 ∈ L∞ (). We will write JQ (U1 )JQ (U2 ), Q
Q
Q
Q
if and only if J 1 (U1 ) J 1 (U2 ) and J 2 (U1 ) J 2 (U2 ) where u1 u2 , U2 = , U1 = v1 v2
ju + jx u Q1 (u, v) Q = . = v Q2 (u, v) − We will first prove that the resolvent JQ is order preserving in the previously defined sense: suppose U1 U2 , i.e., u1 u2 and v1 v2 , then u1 − u2 0,
v1 − v2 0,
i.e., (u1 − u2 )+ = 0,
(v1 − v2 )+ = 0.
Therefore, by (38) we have (JQ (U1 ) − JQ (U2 ))+ 1 Q Q Q Q = {|(J 1 (U1 ) − J 1 (U2 ))+ | + |(J 2 (U1 ) − J 2 (U2 ))+ |} dx 0,
Q
Q
from which it follows that a.e. on we have J i (U1 ) − J i (U2 ) 0, i = 1, 2, i.e., JQ (U1 ) JQ (U2 ) and JQ is order preserving. We may consider the solution of (33) corresponding to the initial value V, i.e., S Q (.)V , the semigroup motion through V . The resolvent identity, ˜ ˜ + Q)−1 , ˜ + Q)−1 − ( I ˜ + Q)−1 = ( ˜ − )( I + Q)−1 (I (I gives us −1
(I + Q)
− (I + Q)
−1
=
− (I + Q)−1 (I + Q)−1 ,
˜ = 1/ , where we used the notation = 1/, ˜ and the order preservation property that
Q n−1 − Q n (J ) V + (J ) V (JQ )n V = JQ ((JQ )n−1 V ) = J Q J Q ((JQ )n V ),
a.e. on , ∀ , > 0, n ∈ N .
P. Kordulová / Nonlinear Analysis: Real World Applications 8 (2007) 1398 – 1409
1409
Applying this estimate with = t/n and passing to the limit as n → ∞ yields S Q (t)V J Q S Q (t)V ,
a.e. on , ∀t > 0, > 0.
If we iterate this last inequality n times, we obtain for = s/n, Q n Q ) S (t)V , S Q (t)V (Js/n
a.e. on , ∀t, s > 0,
and thus, in the limit (n → ∞), S Q (t)V S Q (s)S Q (t)V , =S Q (t + s)V ,
a.e. on , ∀t, s > 0.
(39)
Furthermore, because JQ is order preserving, V U0 0
implies JQ V JQ U0 JQ 0,
and also S Q (t)V S Q (t)U0 S Q (t)0 =
0 v0 (x)
0.
(40)
The last estimate together with monotonicity in (39) implies that V∞ = .1 − limt→∞ S Q (t)V exists and V∞ ∈ Q−1 0. However, as (40) gives us, 0 S Q (t)U0 S Q (t)V ,
a.e. on , ∀t > 0,
it follows that S Q (t)U0 1 exists and S Q (t)u → 0, S Q (t)v → v(x) as t → ∞, v0 (x) v(x).
References [1] V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff International Publishing, Leiden, 1976. [2] M. Brokate, J. Sprekels, Hysteresis and Phase Transitions, Springer, Berlin, 1996. [3] M.G. Crandall, An introduction to evolution governed by accretive operators in dynamical systems, Proceedings of the International Symposium, vol. I, Brown Univ. Providence, RI, 1974, Academic Press, New York, 1976, pp. 131–156. [4] J. Kopfová, Semigroup approach to the question of stability for a partial differential equation with hysteresis, J. Math. Anal. Appl. 223 (1998) 272–287. [5] M.H. Pavel, Nonlinear Evolution Operators and Semigroups, Springer, Berlin, 1987. [7] H.K. Rhee, R. Aris, N.R. Amundson, First order partial differential equations, Theory and Applications of Single Equations, vol. 1, Prentice-Hall, Englewood Cliffs, NJ, 1986. [8] D.M. Ruthen, Principles of Adsorption and Adsorption Processes, Wiley, New York, 1984. [9] R.E. Showalter, M. Peszynska, A transport model with adsorption hysteresis, Differential Integ. Equations 11 (1998) 327–340. [10] A. Visintin, Differential Models of Hysteresis, Springer, Berlin, 1995. [11] P. Wittbold, Asymptotic behavior of certain nonlinear evolution equation in L1 , in: Progress in Partial Differential Equations: The Metz Surveys, vol. 2, 1992, pp. 216–230, Longman Sci. Tech., Harlow, 1993 (Pitman Res. Notes Math. Ser. vol. 296).
Asymptotic behaviour of a quasilinear hyperbolic equation with hysteresis夡 P. Kordulová∗ Mathematical Institute, Silesian University in Opava, 746 01 Opava, Czech Republic Received 31 May 2006; accepted 19 July 2006
Abstract The paper deals with the asymptotic behaviour of the solution of a quasilinear hyperbolic equation with hysteresis. A stability result for solution in L1 () is derived by the nonlinear semigroup approach. 䉷 2006 Elsevier Ltd. All rights reserved. MSC: 34C55; 47J40; 35L60 Keywords: Hysteresis operator; Asymptotic behaviour
1. Introduction We study the asymptotic stability of the equation j(u + v) ju + =0 jt jx u(0, t) = 0,
in (0, L) × [0, T ], (1)
where v = Fr [u] exhibits hysteresis of a type of a generalized play. Hysteresis is represented by functional describing adsorption and desorption on the particles of substance. We introduce here a generalized play operator as a model for adsorption–desorption [8,9]. The motivation comes from applications in chemical and geological engineering; see [7,8]. The purpose of this work is to study asymptotic behaviour of the solution. Firstly, we transform our equation with hysteresis into a system of differential inclusions containing an accretive operator. We investigate the solution based on the theory of nonlinear semigroups, which is used for solving equations with hysteresis [1,3,5,10]. We devote a large part of the paper to a proof of m- and T-accretivity of the operator provided that Fr [.] is generalized play operator, the existence and uniqueness of integral solution for (1) follows. It was proved that its unique integral solution fulfils the entropy condition. In order to get the stability result we apply the theorem of Wittbold [4,11]. 夡 The research was supported, in part, by project GD 201/03/H152 from the Czech Science Foundation and MSM4781305904 from the Czech Ministry of Education. ∗ Fax: +420 553684680. E-mail address: [email protected].
1468-1218/$ - see front matter 䉷 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.nonrwa.2006.07.007
P. Kordulová / Nonlinear Analysis: Real World Applications 8 (2007) 1398 – 1409
1399
These results can be extended to possibly discontinuous generalized Prandtl–Ishlinskii operators of play type. This includes the case of possibly discontinuous Preisach operators. The paper is organized as follows. In Section 2 the concept of hysteresis is introduced. We define the generalized play operator and describe its hysteresis behaviour; see [2,10]. This definition of the generalized play operator includes discontinuous relay operator. We give definitions of a stationary supersolution and a stationary subsolution and recall the asymptotic result of Wittbold in Section 3. Section 4 is devoted to a semigroup approach and contains important theorem about m- and T-accretivity of the operator together with its proof. Moreover, a theorem about existence and uniqueness of integral solutions [10] is recalled. In Section 5 we apply the theorem of Wittbold together with the theorem about accretivity for hyperbolic operators with hysteresis to get the asymptotic behaviour of the solution. 2. Hysteresis play operator Hysteresis is a phenomenon in which the response of a physical system to an external influence depends not only on the present magnitude of the influence but also on the previous history of the system. Hysteresis operators are characterized by two main properties—memory effect and rate independence. In this section we recall definition of the generalized play operator [2,10]. Formally, let l , r : R → P( R) be maximal monotone (possibly multivalued) functions with inf r (u) sup l (u)
∀u ∈ R.
(2)
We set R := [−∞, +∞]. The corresponding input–output behaviour is the following (see Fig. 1): when u is increasing and a given initial value v0 (x) ∈ r (u), v follows the right hysteresis boundary curve. When u starts decreasing, we will move inside the hysteresis loop and v(t) is constant till the moment when u hits the left hysteresis boundary curve and u decreases on it. If u is increasing again, we move inside the loop again, etc. Let v0 ∈ R is given, u be a continuous piecewise linear function [0, T ] → R (namely, function whose graph is a polygonal). For any t ∈ [0, T ], let t0 = 0 < t1 < · · · < tN = T be such that u is linear (or, more precisely, affine) in [ti−1 , ti ] for i = 1, 2, . . .. Then we set recursively if t = 0, min{l (u(0)), max{r (u(0)), v0 }} Fr (t) = min{l (u(t)), max{r (u(t)), v(ti−1 )}} if t ∈ (ti−1 , ti ], i = 1, 2, . . . . Fr is called generalized play (Fig. 1). The hysteresis relation is assumed to hold pointwise in space, v(x, t) = [Fr (u(x, .), v0 (x))](t) in [0, T ] a.e. in ,
v
γl
γr
Fig. 1. Generalized play.
u
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P. Kordulová / Nonlinear Analysis: Real World Applications 8 (2007) 1398 – 1409
where Fr is a hysteresis operator, v0 (x) a given initial value, let be an open bounded subset of RN (N 1) of Lipschitz class. 3. The sub/supersolution method In this chapter we recall results about the asymptotic behaviour of the equation ut + Au 0,
u(0) = u0 ,
(3)
where let A be an m- and T-accretive operator in L1 (). Definition 3.1. A stationary supersolution of (3) is defined to be a function v ∈ L1 () satisfying u0 v,
a.e. on ,
(4)
and (I + A)−1 v v,
a.e. on , ∀ > 0.
(5)
A stationary subsolution of (3) is defined in the same way with reversed inequalities. Remark 3.1. Note that if v ∈ D(A) and if A is single valued, then (4) and (5) are equivalent to v u0 , a.e. on , Av 0, a.e. on .
(6) (7)
Also note that if A is an accretive operator, (6) and (7) imply (4) and (5). Theorem 3.1 (Wittbold [11]). Let A be an m- and T-accretive operator in L1 (), i.e., R(I + A) = L1 (),
∀ 0,
(8)
and ˜ + 1 , (u − u) ˜ + 1 , (JA (u) − JA (u))
∀0, u, u˜ ∈ L1 (),
(9)
where JA (u) = (I + A)−1 (u),
and R denotes the range.
(10)
Suppose that A−1 0 = {0} and let u0 ∈ D(A). Then the following holds: if there exist a stationary subsolution and a stationary supersolution of ut + Au 0, u(0) = u0 , in L1 (), then the solution u of (3) is stable, i.e., u(x, t)1 → 0 as t → ∞. The existence of sub/supersolutions alone implies stability in L1 (). In particular, we do not need the full strength of accretivity of operators to apply this method, but only that resolvents are order preserving. 4. Semigroup approach Here we study the accretivity properties of the generalized play operator for equation (1) in × [0, T ], where is represented by (0, L). We denote by R21 the Banach space of vectors U := (u, v) ∈ R2 , endowed with the norm (u, v)R2 := |u| + |v| ∀(u, v) ∈ R21 . 1
P. Kordulová / Nonlinear Analysis: Real World Applications 8 (2007) 1398 – 1409
L1 (, R21 ) is a Banach space endowed with the norm U L1 (,R2 ) := (|u(x)| + |v(x)|) dx 1
∀ U := (u, v) ∈ L1 (, R21 ).
1401
(11)
The space H is defined as H := {u ∈ H 1 (), u(0) = 0 in the sense of traces}. In the space L2 (, R21 ) the norm is denoted by .2 and the scalar product by ., . . The norm in H is denoted by .H Eq. (1) is equivalent to the Cauchy problem: ⎧ ⎨ jU + A(U ) + R(U ) 0, in × [0, T ] jt (12) ⎩ U (0) = U0 , where D(A) := {U := (u, v) ∈ R2 : inf r (u) v sup l (u)}, A(U ) := {(, −) ∈ R2 : ∈ (U ) ∩ R} B(u) :=
∀U ∈ D(A),
(13)
ju , jx
R(U ) := (B(u), 0), D(R) := {U ∈ L1 (, R21 ) : Bu ∈ L1 ()}, Q(U ) := A(U ) + R(U ), D(Q) := {U := (u, v) ∈ L1 (, R21 ) : U ∈ D(A) a.e. in , u ∈ W 1,1 (), u(0) = 0},
(14)
and by setting U := (u, v), U0 := (u0 , v0 ), ⎧ {+∞} if v < inf r (u), ⎪ ⎪ ⎪ + ⎪ ⎪ if v ∈ r (u)\l (u), R ⎪ ⎪ ⎪ ⎨ {0} if sup r (u) < v < inf l (u), (u, v) = − ⎪ ⎪ if v ∈ l (u)\r (u), R ⎪ ⎪ ⎪ ⎪ ⎪ {−∞} if v > sup l (u), ⎪ ⎩ R if v ∈ l (u) ∩ r (u).
(15)
+ − Here R := [−∞, +∞], R := [0, +∞], R := [−∞, 0].
R) is such Proposition 4.1 (Accretivity. General case). Assume that the (possibly multivalued) function
: R2 → P( that D(
) := {(u, v) ∈ R2 :
(u, v) ∩ R = ∅} = ∅, and
Set
) ∀ (ui , vi ) ∈ D(
if u1 < u2 and v1 > v2
∀ i ∈
(ui , vi )(i = 1, 2), then 1 2 .
(u, v) ∩ R = ∅}, D(Aˆ ) := {U := (u, v) ∈ R2 :
ˆ ) := {(, −) ∈ R2 : ∈
ˆ A(U (u, v) ∩ R} ∀U ∈ D(A).
(16)
(17)
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P. Kordulová / Nonlinear Analysis: Real World Applications 8 (2007) 1398 – 1409
Then Aˆ is T-accretive in R21 . Moreover, if (u, v)} ∃ aˆ > 0 : ∀z ∈ R, Gz : u → {v ∈ R : v − z ∈ aˆ
is a maximal monotone (possibly multivalued) function in R2 ,
(18)
then Aˆ is m-accretive in R21 . Proposition 4.1 is proved in Visintin [10]. Theorem 4.1 (Accretivity. Rate independent case). Assume that l and r fulfil (2) and that D(A) = ∅. Then the operator A is T- and m-accretive in R21 . Proof. Direct application of Proposition 4.1, as condition (18) is fulfilled for any aˆ > 0. In fact, for any (u, z) ∈ R2 and any aˆ > 0, Gz defined in (18) as follows: ⎧ if z inf r (u), Gz (u) = r (u) ⎪ ⎪ ⎪ ⎨ Gz (u) = [z, sup r (u)] if inf r (u) < z < sup r (u), (19) if sup r (u) z inf l (u), Gz (u) = {z} ⎪ ⎪ ⎪ Gz (u) = [inf l (u), z] if inf l (u) < z < sup l (u), ⎩ Gz (u) = l (u) if sup l (u) z; is a maximal monotone (possibly multivalued) function.
Theorem 4.2. Assume that l (u), r (u) fulfil (2) and are affinely bounded, that is, there exist constants C1 , C2 > 0, such that ∀w ∈ R, ∀z ∈ h (w) |z| C1 |w| + C2 ,
(h = l, r).
(20)
Let A and R be defined as previously. Then the operator A + R is m- and T-accretive in L1 (, R21 ). Proof. Visintin [10] just outlined the argument, which is similar to that used for the linear second order elliptic operator in divergence form. The main difference is in the first part of the proof. Here the whole proof is made in detail. We split it into several steps. In the first three steps we study our operator in L2 (, R21 ). In the first step we use Yosida approximation of curves r and l and for any n ∈ N . We prove m-accretivity of the operator Q. In the second one we show T-accretivity of the operator, by using the Heaviside graph, for any n ∈ N . In the next step we take the limit in n and get the same properties also for the limit operator. Finally it is studied in L1 (, R21 ). (1) Let us consider the Yosida approximation of r and l ,
−1
−1 1 1 , ln := n I − I + l ∀n ∈ N , rn := n I − I + r n n and define the corresponding An and n as in (13), (15). Let us fix any n ∈ N. We claim that, if the positive constant a is sufficiently small, then for any F := (f1 , f2 ) ∈ L2 (, R21 ) there exist un ∈ H01 (), vn , n ∈ L2 () such that Bun ∈ L2 () and un + an + aBun = f1 a.e. in , (21) a.e. in , vn − an = f2 n ∈ (un , vn ) a.e. in . To prove this statement, let us define Gn := Gf2 ,n as in (19); this is a single-valued function R → R. By eliminating n and vn, the latter system becomes un + Gn (un ) + aBun = f1 + f2
a.e. in .
(22)
P. Kordulová / Nonlinear Analysis: Real World Applications 8 (2007) 1398 – 1409
The functional a : u¯ →
1403
(u¯ + aBu) ¯ u¯ dx
is not coercive for any a > 0. Therefore, we approximate R by means of R − ε, where ε is a positive parameter and D(R − ε) = D(R) ∩ H. The corresponding functional aε is coercive and a is lower semicontinuous. This yields an equation of the form un + Gn (un ) + a(B − ε)un = f1 + f2
a.e. in .
(23)
In fact, we have u¯ 2 + a [u¯ + a(B − ε)u] ¯ u¯ dx = = u¯ 2 dx +
j2 u¯ ju¯ u¯ − aε 2 u¯ dx jx jx ju¯ 2 j2 u¯ 1 a aε 2 u¯ dx dx − jx 2 jx 2 1 ju¯ = u¯ 2 + aε dx + a[u¯ 2 (x)]L 0 jx 2 2 ju¯ L ju¯ 2 u¯ + aε dx − aε u¯ = jx 0 jx 1 ju¯ + a u¯ 2 (L) − aε (L)u(L). ¯ 2 jx
The first addendum is positive, the last two addendums give a constant c. According to the definition of coercivity we have if u ¯ H → ∞: 2 2 + ju¯ + c u ¯ jx 2 u¯ + aBu, ¯ u
¯ 2 = lim lim u ¯ H →∞ u ¯ H →∞ u ¯ H u ¯ H u ¯ 2H = lim = lim u ¯ H = ∞. u ¯ H →∞ u ¯ H →∞ ¯ H u Hence by standard results there exists a unique solution un ∈ H of (23). We take the limit as ε → 0 in (23) and from it we get the existence and uniqueness of a solution of (22). Then vn = Gn (un ), and n is determined by the second equation of (21). (2) We claim that for any n ∈ N and for any F1 , F2 ∈ L2 (, R21 ), setting Ui := (I + aAn + aR)−1 (Fi ) (i = 1, 2), (U1 − U2 )+ L1 (,R2 ) (F1 − F2 )+ L1 (,R2 ) . 1
1
(24)
Then the analogous inequality holds for the negative parts, whence (U1 − U2 )L1 (,R2 ) (F1 − F2 )L1 (,R2 ) . 1
1
(25)
In order to prove (24), first let us take any > 0, set ⎧ ⎪ ⎨ 0u¯ if u¯ < 0, ¯ = H (u) if 0 u¯ , ∀u¯ ∈ R, ⎪ ⎩ 1 if u¯ > , and denote by H the Heaviside graph. In this argument we omit the (fixed) index n. For i = 1, 2, let us denote by (ui , vi , i ) the solution of Eq. (21) corresponding to Fi and set U˜ = (u, ˜ v) ˜ := U1 − U2 , ˜ := 1 − 2 .
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P. Kordulová / Nonlinear Analysis: Real World Applications 8 (2007) 1398 – 1409
For any measurable function h1 such that h1 ∈ H (v) ˜ a.e. in , by (21) we have ˜ + ] dx (F1 − F2 )+ L1 (,R2 ) = [(u˜ + a ˜ + aBu) ˜ + + (v˜ − a ) 1 ˜ 1 ] dx [(u˜ + a ˜ + aBu)H ˜ (u) ˜ + (v˜ − a )h ˜ (u) ˜ − h1 ]} dx {uH ˜ (u) ˜ + vh ˜ 1 + a [H +a ˜ dx. BuH ˜ (u)
Note that
H (u) ˜ → h0 = Hence
0 1
if u˜ 0 if u˜ > 0
(26)
a.e. in .
[uH ˜ (u) ˜ + vh ˜ 1 ] dx → (u˜ + + v˜ + ) dx = U˜ + L1 (,R2 ) , 1 ˜ 0 − h1 ) dx. ˜ (u) (h [H ˜ − h1 ] dx →
Since rn , ln are single-valued, then ∀u, v1 , v2 ∈ R, if v1 < v2 , then ∀1 ∈
(u, v1 ), ∀2 ∈
(u, v2 ), 1 2 holds. Then we have ∃h1 ∈ H (v) ˜ : ∀h0 ∈ H (u), ˜
˜ 0 − h1 ) 0 a.e. in . (h
Hence, for this choice of h1 , the limit as → 0 of the first integral of the last member of (26) is not smaller than U˜ + L1 (,R2 ) . 1 u¯ Now let us consider the last integral of (26). To deal with the first order term, let us set (u) ¯ := 0 H () d for any u¯ ∈ R; thus ¯ 0 (u)(1/)
d = /2.
0
So we have ju˜ ˜ j ju H (u) dx = − (u) ˜ dx = − uH ˜ ˜ dx 0. jx jx jx Hence we conclude that lim inf →0 BuH ˜ (u) ˜ dx 0. Therefore, (24) holds. (3) Let us now take the limit in n. Multiplying (23) by un and integrating in we get 2 ju¯ 2 un + Gn (un )un + aε + c dx f1 + f2 L2 () un H jx un 2H − c3 un H − c2 f1 + f2 L2 () un H c2 un H − c3 − f1 + f2 L2 () . un H Now we have two cases: (a) un H is unbounded. It is impossible due to assumption of f1 + f2 , (b) un H is bounded. So that un H constant (independently of n).
(27)
P. Kordulová / Nonlinear Analysis: Real World Applications 8 (2007) 1398 – 1409
1405
So it is easy to see that un H ,
vn L2 () constant (independently of n).
Hence there exist u, v ∈ L2 () such that, possibly taking n → ∞ along a subsequence, un → u weakly in H,
(28)
vn → v
(29)
weakly in L2 ().
u¯ Setting n (u) ¯ := 0 Gn (y) dy for any u¯ ∈ R, (22) is equivalent to the following variational inequality: [(un + aBun − f1 − f2 )(un − v) + n (un ) − n (v)] dx 0
∀v ∈ L2 ().
(30)
Let us denote by a lower semicontinuous convex function such that j = Gf2 (j denoting the subdifferential of
). Because of the lower semicontinuity of , u solves the limit inequality, obtained from (30) by replacing n by . We employ here the compactness of the inclusion H ⊂ L2 (). We take the limit as ε → 0 in (23). Hence we can pass to the limit in (22), getting v := −u − aBu + f1 + f2 ∈ Gf2 (u)
a.e. in .
Therefore, setting := (1/a)(v − f2 ) ∈ L2 (), we get
u + a + aBu = f1 v − a = f2 ∈ (u, v)
a.e. in , a.e. in , a.e. in .
(31)
Note that, by comparison of the terms of (31)1 , Bu ∈ L2 (), and this equation holds a.e. in . Also for this limit problem the solution is unique. When we take the limit in (24) and (25), we get the T-accretivity also for the limit operator A + R. (4) Finally, we have to study our operator in L1 (, R21 ). Let us fix any F ∈ L1 (, R21 ), any sequence {Fm ∈ 2 L (, R21 )} such that Fm → F
strongly in L1 (, R21 ),
and set Um := (I + aA + aR)−1 (Fm )(∈ L2 (, R21 )). By (25), there exists U ∈ L1 (, R21 ) such that Um := (um , vm ) → U := (u, v)
strongly in L1 (, R21 );
this yields vm − f2m v − f2 → =: strongly in L1 (), a a aBum = f1m − um − am → f1 − u − a = aBu strongly in L1 (). m =
Therefore, (u, v, ) solves the corresponding problem (31). Then we can pass to the limit in the inequalities (24) and (25), provided that we show that the solution of (31) is unique. To this aim, let us fix any f1 , f2 ∈ L1 (), consider two solutions (ui , vi , i ) (i = 1, 2), take the difference between the system (31) written for i = 1, 2, and set u˜ := u1 − u2 , v˜ := v1 − v2 . By eliminating 1 and 2 , we get u˜ + v˜ + aBu˜ = 0 a.e. in , (32) vi ∈ Gf2 (ui )(i = 1, 2) a.e. in .
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P. Kordulová / Nonlinear Analysis: Real World Applications 8 (2007) 1398 – 1409
Now let us multiply the first equation by any measurable function s such that s ∈ sign(u˜ + v) ˜ a.e. in . Since Gf2 is nondecreasing, we have s ∈ sign(u); ˜ hence, setting sj (y) := arctan(jy) for any j ∈ N and any y ∈ R, we get ˜ dx = −a lim (Bu)s ˜ j (u) ˜ dx. |u˜ + v| ˜ dx = −a (Bu)s
j →∞
By a calculation similar to (27), we get that the latter limit is nonnegative. This yields the desired uniqueness property. As we saw, the operator occurring in the Cauchy problem is m- and T-accretive. Here we apply some classical results of the theory of nonlinear semigroups (Theorem 4.3). Theorem 4.3. Let B be a Banach space and Z : D(Z) ⊂ B → B be an m-accretive operator. Then (i) If u0 ∈ D(Z), then Cauchy problem has one and only one integral solution. (ii) The integral solution depends Lipschitz continuously on the data u0 . (iii) If u0 ∈ D(Z), then the integral solution of Cauchy problem is Lipschitz continuous in [0, T ]. Theorem 4.4. Let be an open subset of R of Lipschitz class. Let L1 (, R21 ) be endowed with the norm (11). Define the operator R as in (14). Let A be as above, and assume that (20) holds. Take any U0 := (u0 , v0 ) ∈ L1 (, R21 ) such that U0 ∈ D() a.e. in , and set Q := A + R. Then the Cauchy problem (12) has one and only one integral solution U : [0, T ] → L1 (, R21 ), which depends continuously on data u0 , v0 . Moreover, if Ru0 ∈ L1 (), then U is Lipschitz continuous. 5. Main result In this section we prove two new lemmas and apply the above results to get the asymptotic behaviour of the solution, the main result of the paper. Theorem 5.1. Suppose all conditions of Theorem 4.2 are satisfied, i.e., the operator A + R is m- and T-accretive in L1 (, R21 ). Suppose also that u0 ∈ D(Q). Then there exists v∞ (x) dependent on x only, such that for the solution u U= v of jU + Q(U ) = 0, jt u0 U (0) = , v0 the following holds: .1 − lim u(x, t) = 0, t→∞
.1 − lim v(x, t) = v∞ (x). t→∞
Before proving this theorem we will prove two lemmas which will be used in the proof of the theorem. Lemma 5.1. Let Q be m- and T-accretive operator in L1 (, R21 ). Then the following holds:
0 Q−1 0 = , such that inf r (0)v(x) sup l (0) . v(x)
(33) (34)
P. Kordulová / Nonlinear Analysis: Real World Applications 8 (2007) 1398 – 1409
1407
Proof.
ju + jx u = ; ∈ (U ) ∩ R , v − u so that v ∈ Q−1 0 if and only if the following is satisfied: Q
ju = 0, jx − = 0,
+
which implies = 0, which is satisfied by any v(x), if inf r (u(x))v(x) sup l (u(x)). We then must have ju = 0 on , jx u(0, t) = 0. Therefore, u = 0 is the unique solution of the previous (preceding) equation, and the assertion follows.
Lemma 5.2. If u0 ∈ L∞ (), there exist a stationary supersolution and a stationary subsolution of the equation jU/jt + Q(U ) 0. Moreover, those can be chosen so that they belong to D(Q). Proof. We show only the existence of a stationary supersolution. The proof of existence of a subsolution is analogous. We can look for solutions of +
ju 0, jx
a.e. on ,
−0,
a.e. on ,
u u0 ,
a.e. on ,
v v0 ,
a.e. on .
(35)
We can choose = 0. Then from the first equation and the third one we require ju 0, jx
a.e. on ,
(36)
u u0 ,
a.e. on .
(37)
On the assumption that u0 ∈ L∞ (), u can be easily found, in fact u = u0 L∞ satisfies ju = 0, jx
a.e. on ,
u u0 , a.e. on . u Then v will be the supersolution which can be found if we choose v such that inf r (u) < v < sup l (u). The last assertion in the lemma is obvious. Proof of the main Theorem. Suppose first that all conditions of Theorem 5.1 are satisfied, i.e., the operator Q is mand T-accretive in L1 (, R21 ), R(I + Q) = L1 (, R21 ),
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P. Kordulová / Nonlinear Analysis: Real World Applications 8 (2007) 1398 – 1409
and (JQ (U ) − JQ (U˜ ))+ 1 (U − U˜ )+ 1 ,
(38)
where JQ (U ) = (I + Q)−1 (U ) and
U 1 =
(|u(x)| + |v(x)|) dx,
denotes the norm in L1 (, R21 ). Furthermore, we suppose that u0 ∈ L∞ (). We will write JQ (U1 )JQ (U2 ), Q
Q
Q
Q
if and only if J 1 (U1 ) J 1 (U2 ) and J 2 (U1 ) J 2 (U2 ) where u1 u2 , U2 = , U1 = v1 v2
ju + jx u Q1 (u, v) Q = . = v Q2 (u, v) − We will first prove that the resolvent JQ is order preserving in the previously defined sense: suppose U1 U2 , i.e., u1 u2 and v1 v2 , then u1 − u2 0,
v1 − v2 0,
i.e., (u1 − u2 )+ = 0,
(v1 − v2 )+ = 0.
Therefore, by (38) we have (JQ (U1 ) − JQ (U2 ))+ 1 Q Q Q Q = {|(J 1 (U1 ) − J 1 (U2 ))+ | + |(J 2 (U1 ) − J 2 (U2 ))+ |} dx 0,
Q
Q
from which it follows that a.e. on we have J i (U1 ) − J i (U2 ) 0, i = 1, 2, i.e., JQ (U1 ) JQ (U2 ) and JQ is order preserving. We may consider the solution of (33) corresponding to the initial value V, i.e., S Q (.)V , the semigroup motion through V . The resolvent identity, ˜ ˜ + Q)−1 , ˜ + Q)−1 − ( I ˜ + Q)−1 = ( ˜ − )( I + Q)−1 (I (I gives us −1
(I + Q)
− (I + Q)
−1
=
− (I + Q)−1 (I + Q)−1 ,
˜ = 1/ , where we used the notation = 1/, ˜ and the order preservation property that
Q n−1 − Q n (J ) V + (J ) V (JQ )n V = JQ ((JQ )n−1 V ) = J Q J Q ((JQ )n V ),
a.e. on , ∀ , > 0, n ∈ N .
P. Kordulová / Nonlinear Analysis: Real World Applications 8 (2007) 1398 – 1409
1409
Applying this estimate with = t/n and passing to the limit as n → ∞ yields S Q (t)V J Q S Q (t)V ,
a.e. on , ∀t > 0, > 0.
If we iterate this last inequality n times, we obtain for = s/n, Q n Q ) S (t)V , S Q (t)V (Js/n
a.e. on , ∀t, s > 0,
and thus, in the limit (n → ∞), S Q (t)V S Q (s)S Q (t)V , =S Q (t + s)V ,
a.e. on , ∀t, s > 0.
(39)
Furthermore, because JQ is order preserving, V U0 0
implies JQ V JQ U0 JQ 0,
and also S Q (t)V S Q (t)U0 S Q (t)0 =
0 v0 (x)
0.
(40)
The last estimate together with monotonicity in (39) implies that V∞ = .1 − limt→∞ S Q (t)V exists and V∞ ∈ Q−1 0. However, as (40) gives us, 0 S Q (t)U0 S Q (t)V ,
a.e. on , ∀t > 0,
it follows that S Q (t)U0 1 exists and S Q (t)u → 0, S Q (t)v → v(x) as t → ∞, v0 (x) v(x).
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