Asymptotic Behavior of Solutions to Reaction-Diffusion Equations with Dynamic Boundary Conditions and Irregular Data.

1. Introduction

We consider the asymptotic behavior of solutions to the following parabolic equations with dynamic boundary conditions and irregular data,

[mathematical expression not reproducible], (1)

where [OMEGA] is a bounded domain in [R.sup.N](N [greater than or equal to] 3) with smooth boundary [GAMMA]. The first equation is the standard reactiondiffusion equation, and the second equation is the boundary equation, in which the value of u is assumed to be the trace of the function u defined for x [member of] [OMEGA], [[DELTA].sub,[GAMMA]]u is the Laplace-Beltrami operator on r , [mu] > 0 plays the role of a surface diffusion coefficient, and v is the outward normal to [OMEGA]. By irregular data, we mean that [u.sub.0](x), h(x) [member of] [L.sup.1]([OMEGA]), and g(x) [member of] [L.sup.1](r). We also assume that f is [C.sup.0] and there exist positive constants l, [C.sub.1], [C.sub.2], [c.sub.0] > 0 and r [greater than or equal to] 2 such that

[C.sub.1] [[absolute value of s].sup.r] - [c.sub.0] [less than or equal to] f(s) s [less than or equal to] [C.sub.2] [[absolute value of s].sup.r] + [c.sub.0] [for all]s [member of] R (2)

(f (x, [s.sub.1])-f (x, [s.sub.2])) ([s.sub.1] - [s.sub.2]) [greater than or equal to] - l - [[absolute value of ([s.sub.1] - [s.sub.2])].sup.2] [for all][s.sub.1], [s.sub.2] [member of] R (3)

Partial differential equations with dynamic boundary conditions like (1) have applications in various fields such as hydrodynamics, heat transfer theory, and thermoelasticity [2-5]. The existence and uniqueness of solutions for problem (1) have been studied extensively in various contexts; see, e.g., [2, 6-9]. The long time behavior of solutions to (1) and related models have also aroused much interests in recent years. For autonomous equations, in  the existence of global attractors was derived under the assumption [mu] = 0, g,h = 0. Then in [11-13] the existence of global attractors and their fractal dimensions were further studied for certain semilinear reaction-diffusion equations with dynamic boundary conditions, while in [9, 14, 15], the existence of global attractors was obtained for more general quasilinear parabolic equations with dynamic boundary conditions. For the nonautonomous case, the existence of pullback attractors for parabolic equations with dynamic boundary conditions was first obtained in , and then in [17-19], while the existence of uniform attractors for linear and quasilinear parabolic equations with dynamic boundary conditions was investigated in [12,20].

In the references aforementioned, the initial data and forcing terms involved are mostly assumed to be regular (belonging to [L.sup.2]([OMEGA]) or even [L.sup.[infinity]]), and few results were known when the initial data and forcing terms are not regular, such as [L.sup.1] functions . This motivates us to investigate the existence and uniqueness results as well as the large-time behavior of solutions to problem (1) with [L.sup.1] data. Due to irregular initial data and forcing terms, the usual framework for the existence and uniqueness of solutions does not work here. Also the large-time behavior for parabolic equations with [L.sup.1] data is much more involved. The less regularity of the data influences the regularity of the solutions greatly, which in turn causes some crucial difficulties in investigating the asymptotic behaviors of solutions.

In this paper, to derive the existence and uniqueness result we shall work in the framework of entropy solutions, which was first introduced in  for elliptic equations involving measure data and was then adapted to parabolic equations with [L.sup.1] data in . We will borrow some ideas in [23, 24] and use smooth approximations to derive the existence of the entropy solution, whereas, to cope with the dynamic boundary conditions, some delicate analysis must be addressed. For the large-time behavior of the entropy solution, we prove the existence of global attractor in [L.sup.1]([bar.[OMEGA]],dv). It is well known that to obtain global attractors, the most essential step is to derive the compactness of the semigroup, which more or less relies on certain uniform estimates in higher order spaces. Here to overcome the difficulties brought by the irregular initial data and forcing terms, we will perform some delicate Marcinkiewicz estimates on the solution and use the Aubin-Simon type compactness results to derive the compactness of the solution semigroup.

We mention that the existence and uniqueness results for elliptic or parabolic equations with Dirichlet boundary condition and [L.sup.1] or measure data have been studied extensively in the past years; see [24-26] and large amount of references therein. The large-time behaviors for parabolic equations with Dirichlet boundary conditions involving irregular data have also been studied by many authors; see, e.g., [27-29]. The results obtained here might be viewed as an extension of the results therein to problems with more general boundary conditions.

The rest of this paper is organized as follows. In Section 2, we provide some preliminaries and the main results of this paper. Then in Sections 3 and 4, we provide the proof for the main results. For convenience, in the following we use [c.sub.i], C, [C.sub.i] to denote generic constants in various occasions, and we will denote [OMEGA] x (0, T), [GAMMA] x (0, T) as [Q.sub.T] and [[GAMMA].sub.T], respectively. For a function u on [Q.sub.T] and [[GAMMA].sub.T], we set {[OMEGA](u [greater than or equal to] k)} = {(x, t) [member of] [Q.sub.T] : u(x,t) [greater than or equal to] k}, {[GAMMA](u [greater than or equal to] k)} = {(x,t) [member of] [[GAMMA].sub.T] : u(x,t) [greater than or equal to] k}.

2. Preliminaries and the Main Result

To deal with the dynamic boundary conditions, we introduce the Lebesgue spaces as follows (see  for more details). Let [OMEGA] [subset] [R.sup.N] be a bounded domain with smooth boundary [GAMMA]r. For 1 [less than or equal to] p < 1, define the Lebesgue space [L.sup.P]([bar.[OMEGA]], dv) as

[L.sup.P] ([bar.[OMEGA]], dv) = {U = (u, [upsilon]):u [member of] [L.sup.P] ([OMEGA]), [upsilon] [member of] [L.sup.P] ([GAMMA])}, (4)

with norm

[mathematical expression not reproducible] (5)

where dv = dx|[sub,[OMEGA] [direct sum] dS|[sub.[GAMMA]] on [bar.[OMEGA]] is defined by v(A) = [absolute value of A [intersection] [omega]] + S(A [intersection] [GAMMA]) for any measurable set A [subset] [bar.[OMEGA]]. Define the Sobolev space [H.sup.]1([bar.[OMEGA]], dv) as

[H.sup.1] ([bar.[OMEGA]], dv) = {U := (u, [upsilon]):u [member of] [H.sup.1] ([OMEGA]), [upsilon] [member of] [H.sup.1] ([GAMMA])} (6)

with norm

[mathematical expression not reproducible]. (7)

It is easy to see that we can identify [H.sup.1]([bar.[OMEGA]], dv) with [H.sup.1]([OMEGA]) [direct sum] [H.sup.1]([GAMMA]) under this norm. Hereafter, we denote by [H.sup.-1]([[bar.[OMEGA]], dv) the dual space of [H.sup.1]([bar.[OMEGA]], dv).

Let [[phi].sub.k] be the truncation at height k [greater than or equal to] 0,

[mathematical expression not reproducible], (8)

and denote [[PSI].sub.k] : R [??] [R.sup.+] as its primitive function, i.e.,

[mathematical expression not reproducible]. (9)

It is obvious that [[PSI].sub.k](s) [greater than or equal to] 0 and [[PSI].sub.k](s) [less than or equal to] k[absolute value of s].

We work within the framework of entropy solutions defined as follows.

Definition 1. A function u is called an entropy solution of problem (1), if for any T > 0 and k > 0, u [member of] C(0, T; [L.sup.1])([bar.[OMEGA], dv)) [intersection] [L.sup.r-1] (0, T; [L.sup.r-1]([bar.[OMEGA]];dv)); moreover,

[mathematical expression not reproducible](10)

for all k > 0 and u [member of] C(0, T; [L.sup.2]([bar.[OMEGA]], dv)) [intersection] [L.sup.[infinity]](0, T; [L.sup.[infinity]]([bar.[OMEGA]]; dv)) such that [[phi].sub.t] [member of] [L.sup.2](0, T; [H.sup.-1] ([bar.[OMEGA]], dv)).

Our first result concerns on the existence and uniqueness of entropy solutions.

Theorem 2. Under assumptions (2)-(3) and suppose that [u.sub.0],h [member of] [L.sup.1]([member of]) and g [member of] [L.sup.1]([GAMMA]), there exists a unique entropy solution for problem (1).

To give the second result on the long time behavior of solutions, we recall the definition of global attractors.

Definition 3 (see ). Let [{S(t)}.sub.t[greater than or equal to]0] be a semigroup on a Banach space X. A subset A [subset] X is called a global attractor for the semigroup if A is compact in X and enjoy the following properties:

(i) A is invariant, i.e., S(t)A = A for any t [greater than or equal to] 0;

(ii) A attracts every bounded subset of X, i.e., for any bounded subset B of X and any neighborhood N(A) of the set A, there exists a [T.sub.0] = [T.sub.0](B, N(A)) such that

S(t)B [subset] N (A) for t [greater than or equal to] [T.sub.0]. (11)

Theorem 4. Assume that [u.sub.0],h [member of] [L.sup.1]([OMEGA]), g [member of] [L.sup.1]([GAMMA]) and f satisfies assumptions (2)-(3); then the semigroup generated by problem (1) admits a global attractor A in [L.sup.1]([bar.[OMEGA]], dv); i.e., A is compact, invariant in [L.sup.1] ([bar.[OMEGA]], dv) and attracts every bounded subset of [L.sup.1]([bar.[OMEGA]], dv) in the norm topology of [L.sup.1]{[bar.[OMEGA]], dv).

To prove the theorems above, let us first provide some preliminaries.

For s [member of] R, define the Marcinkiewicz space [M.sup.s]([bar.[OMEGA]], dv) as the set of measurable functions v such that

[mathematical expression not reproducible], (12)

for some positive constant C and all k > 0. We have the following.

Lemma 5. Let r, s be positive constants such that r - s > 0 and let u(x, t) be a function defined on ([Q.sub.T], dv). If u [member of] [M.sup.r]([Q.sub.T], dv), then [[absolute value of u].sup.s] [member of] [L.sup.1]([Q.sub.T],dv). In particular, [M.sup.r]([Q.sub.T],dv) [subset] [L.sup.s]([Q.sub.T], dv) for all s,r [greater than or equal to] 1 such that r - s > 0.

The following is the well-known Aubin-Simon compactness result.

Lemma 6 (see ). Assume X [subset] Z [subset] Y with compact imbedding X [??] Z (X, Y and Z are Banach spaces). Let F be bounded in [L.sup.p](0, T;X), where 1 [less than or equal to] p < [infinity], and [partial derivative]F/[partial derivative]t = {[partial derivative]f/[partial derivative]t : f [member of] F} be bounded in [L.sup.1] (0, T;Y). Then F is relatively compact in [L.sup.p] (0, T; Z).

3. Existence and Uniqueness of Entropy Solutions

In this section, we provide the proof for Theorem 2. We begin with the existence and uniqueness results for the problem with regular data.

Definition 7. Assume that [u.sub.0],h(x) [member of] [L.sup.2]([OMEGA]), g(x) [member of] [L.sup.2]([GAMMA]) and f [member of] [C.sup.0] satisfies (2)-(3). A function u is called a weak solution of problem (1), if for any T > 0, and u [member of] C(0, T;[L.sup.2]([bar.[OMEGA]], dv)) [intersection] [L.sup.r](0, T; [L.sup.r]([bar.[OMEGA]], dv)) [intersection] [L.sup.2](0, T; [H.sup.1] ([bar.[OMEGA]], dv)), and moreover

[mathematical expression not reproducible], (13)

for all [phi] [member of] [C.sup.[infinity].sub.c] ([bar.[OMEGA]].

Theorem 8. Assume that [u.sub.0],h(x) [member of] [L.sup.2]([OMEGA].), g(x) [member of] [L.sup.2]([GAMMA]), f [member of] [C.sup.0] satisfies (2)-(3). Then problem (1) admits a weak solution u [member of] [L.sup.2]{0,T;[H.sup.1](n,dv)) [intersection] C(0,T; [L.sup.2]([bar.[OMEGA]],dv)) [intersection] [L.sup.r](0,T; [L.sup.r]([bar.[OMEGA]],dv)).

Proof. The proof of this theorem is based on the standard Galerkin approximation method as in ; we thus omit the details for concision.

Proof of Theorem 2. Now provide the proof of the existence and uniqueness of entropy solutions for problem (1). For simplicity, we assume that [mu] = 1. Let {[h.sup.n]} [subset] [L.sup.[infinity]]([OMEGA]), {[g.sup.n]} [subset] [L.sup.[infinity]]([GAMMA]), and [u.sup.n.sub.0] [subset] [L.sup.[infinity]]([OMEGA]) be three sequences of functions strongly convergent, respectively, to h in [L.sup.1] ([OMEGA]), to g in [L.sup.1]([GAMMA]) and to [u.sub.0] in [L.sup.1]([OMEGA]) such that

[mathematical expression not reproducible]. (14)

Let us consider the approximation problem of (1),

[mathematical expression not reproducible], (15)

By virtue of Theorem 8, there is an unique weak solution [u.sup.n] to (15) for each n, with

[mathematical expression not reproducible]. (16)

Next, we shall follow the ideas of  to prove that, up to a subsequence, [u.sup.n] converges to a measurable function u, which is the entropy solution of problem (1). Let us divide the proof into several steps. Hereafter, without indication all the convergence should be understood in the sense of subsequences.

Step 1. [u.sup.n] converges to u in [L.sup.1] (0, T; [L.sup.1]{[bar.[OMEGA]],dv)).

Taking [[phi].sub.k]([u.sup.n]) (k [greater than or equal to] 1) as a test function in (15), we deduce that

[mathematical expression not reproducible]. (17)

Since

[mathematical expression not reproducible] (18)

[mathematical expression not reproducible]. (19)

Note that r [greater than or equal to] 2. From the definition of [[PSI].sub.k](x) we obtain

[mathematical expression not reproducible]. (20)

If we choose k = 1 and taking the above inequality in consideration, we deduce from (19) that

[mathematical expression not reproducible]. (21)

By the standard Gronwall's inequality, we obtain that

[mathematical expression not reproducible]. (22)

Note that

[mathematical expression not reproducible]. (23)

By the definition of [[PSI].sub.k] we have

[mathematical expression not reproducible], (24)

[mathematical expression not reproducible] (25)

Therefore, we get

[mathematical expression not reproducible] (26)

for any t >0.

Furthermore, integrating (19) between 0 and T, it is easy to obtain

[mathematical expression not reproducible]. (27)

Setting [lambda] = 2/r and using (27), we deduce that

[mathematical expression not reproducible]. (28)

Similarly we can obtain

[mathematical expression not reproducible]. (29)

Combining (28) and (29), it yields

[mathematical expression not reproducible], (30)

which implies that [[absolute value of [nabla][u.sup.n]].sup.[lambda]] is bounded in [M.sup.r-1]([Q.sub.T],dv). Hence, we conclude from Lemma 5 that [[absolute value of [nabla][u.sup.n]].sup.[beta]] is bounded in [L.sup.1](0, T; ([bar.[OMEGA], dv)) for 1 < [beta] < 2(r - 1)/r, which implies that [mathematical expression not reproducible] is bounded in [L.sup.1](0, T; ([bar.[OMEGA], dv)) for 1 < [q.sub.0] < 2 (r-l)/r. Therefore, [u.sup.n](t) is bounded in [mathematical expression not reproducible].

Furthermore, we obtain that {[u.sup.n.sub.t]} is bounded in [mathematical expression not reproducible]; using Lemma 6, we know that [u.sup.n](t) is relatively compact in [L.sup.1] (0, T; [L.sup.1] ([bar.[OMEGA], dv)). Thus, up to a subsequence [u.sup.n] convergence to u in [L.sup.1]{0, T; [L.sup.1] ([bar.[OMEGA]], dv)).

Step 2. f([u.sup.n]) converges to f(u) in [L.sup.1](0, T; ([L.sup.1]([bar.[OMEGA]], dv)) for any given T > 0.

By Vitali's convergence theorem, it is enough to prove

[mathematical expression not reproducible]. (31)

Given L > 0, we define

[mathematical expression not reproducible]. (32)

Let {[T.sup.i.sub.L]} be a sequence of real smooth increasing functions with [T.sup.i.sub.L](0) = 0 and [T.sup.i.sub.L](s) [right arrow] [T.sub.L](s) as i [right arrow] +[infinity]. Taking [T.sup.i.sub.L]([u.sup.n]) as a test function in (15), we deduce that for any T > 0

[mathematical expression not reproducible] (33)

where [[THETA].sup.i.sub.L] is the primitive function of [T.sup.i.sub.L]. Note that the first integral is nonnegative; thus discarding it and passing to the limit in i, we obtain that

[mathematical expression not reproducible]. (34)

From (2), we know that when [absolute value of s] [greater than or equal to] [[sigma].sub.0] we have

[c.sub.1] [[absolute value of s].sup.r-1] - C/[[sigma].sub.0] f(s) sgn (s) [less than or equal to] [c.sub.2] [[absolute value of r].sup.r-1] + C/[[sigma].sub.0], (35)

and when [absolute value of s] [less than or equal to] [[sigma].sub.0], we have [absolute value of f] < [c.sub.6] Combining this with (35), we obtain that for any s [member of] R

[mathematical expression not reproducible], (36)

where [C.sub.1] = C/[[sigma].sub.0], [C.sub.2] = max{[c.sub.1], [c.sub.2]}. This implies that

[absolute value of f(s)] [less than or equal to] [C.sub.3] (f(s) sgn (s) + 1) (37)

for some positive constant [C.sub.3]. Thus, we deduce from (34) that

[mathematical expression not reproducible]. (38)

Since for any T > 0, [u.sup.n] converges to u in [L.sup.1](0, T; ([bar.[OMEGA], dv)); there exists a positive constant [C.sub.4] independent of n, such that

[mathematical expression not reproducible]. (39)

Then we have

[mathematical expression not reproducible]. (40)

For any [epsilon] > 0, we can always find a positive constant [L.sub.1] such that

[mathematical expression not reproducible], holds for all n. (41)

Now we analyze each term of the right hand side of (38); note that {[h.sup.n]} converges to h in [L.sup.1] ([OMEGA]), and

[mathematical expression not reproducible]. (42)

For any given [epsilon] >0, the first term on the right hand side can be strictly less then [epsilon] whenever n > [N.sub.1]. Thanks to (41), for L large enough (L > [L.sub.1]), the second term can be strictly less than e for all n (absolute continuity of the Lebesgue integral). Also, we can always find a positive constant [L.sub.2], such that

[mathematical expression not reproducible]. (43)

So setting [L.sub.3] = max{[L.sub.1], [L.sub.2]}), we get

[mathematical expression not reproducible]. (44)

Similarly, setting [L.sub.4] = max{[L.sub.1], [L.sub.2], [L.sub.3]}), we can get

[mathematical expression not reproducible]. (45)

Then for any [epsilon] >0 there exists a [L.sub.5] > 0, such that

[mathematical expression not reproducible]. (46)

Taking (41)-(46) into (38), we obtain that for any [epsilon] > 0 there exists [L.sub.6] = max{[L.sub.1], [L.sub.3], [L.sub.4], [L.sub.5]}, such that

[mathematical expression not reproducible], (47)

uniformly in n. On the other hand, from (36) we have

[mathematical expression not reproducible] (48)

uniformly in n, whenever [absolute value of E] is small enough.

Note that for any E [subset] ([Q.sub.T], dv)

[mathematical expression not reproducible]. (49)

Thus, for any T > 0, f([u.sup.n]) is equi-integrable in ([Q.sub.T], dv) and due to Vitali's convergence theorem, f([u.sup.n]) converges to f(u) in [L.sup.1](0, T; [L.sup.1]([bar.[OMEGA]], dv)).

Step 3. [nabla][u.sup.n] converges to a function [nabla]u in [L.sup.1](0, T; [L.sup.1]([bar.[OMEGA]], dv)).

For [epsilon] >0, taking [phi] = [[phi].sub.[epsilon]]([u.sup.n] - [u.sup.m]) as a test function in (15) one can deduce that

[mathematical expression not reproducible]. (50)

Note that f([u.sup.n]), [h.sup.n], [g.sup.n], and [u.sup.n.sub.0] are convergent in [L.sup.1], and 0 [less than or equal to] [[PSI].sub.[epsilon]](s) [less than or equal to] [epsilon], for 0 < [epsilon] < 1 we obtain that

[mathematical expression not reproducible], (51)

where

[mathematical expression not reproducible]. (52)

Using Holder inequality, we get

[mathematical expression not reproducible], (53)

where [A.sub.1] = [Q.sub.T]\[Q.sup.1.sub.T] = {(x, t) [member of] [Q.sub.T] : [absolute value of ([u.sup.n] - [u.sup.m])] > [epsilon]}. Now let us bound meas ([A.sub.1]); we have

[mathematical expression not reproducible], (54)

[u.sup.n] [right arrow] u in [L.sup.1](0, T; ([bar.[OMEGA]],dv)); hence, we have meas ([A.sub.1]) < [epsilon]. Then we have

[mathematical expression not reproducible]. (55)

In the same way, we have [mathematical expression not reproducible]. So we can obtain that

[mathematical expression not reproducible]. (56)

That is {[nabla][u.sup.n]} is a Cauchy sequence in [L.sup.1](0, T; [L.sup.1]([bar.[OMEGA]], dv)). Discarding the nonnegative term in the left hand side of (50) we can deduce that [u.sup.n] [right arrow] u in C(0,T; [L.sup.1]([bar.[OMEGA]], dv)).

Step 4. Pass to the limits.

Similar to , taking [[phi].sub.k]([u.sup.n] -[phi]) as a test function in (15), we have

[mathematical expression not reproducible], (57)

where [phi] [member of] [L.sup.2](0, T; [w.sup.1,2]([bar.[OMEGA]], dv)) [intersection] [L.sup.[infinity]] (0,T; [L.sup.[infinity]]([bar.[OMEGA]], dv)). Let us study the limit for n [right arrow] [infinity] of each term.

We have seen that [u.sup.n] [right arrow] u in C(0, T; [L.sup.1]([bar.[OMEGA]], dv)); hence, [for all]t [less than or equal to] T, [u.sup.n] [right arrow] u in [L.sup.1]([bar.[OMEGA]], dv). Since [[PSI].sub.k] is k-Lipschitz continuous one has, when n [right arrow] [infinity]

[mathematical expression not reproducible]. (58)

Since

[mathematical expression not reproducible], (59)

and [u.sup.n.sub.0] [right arrow] [u.sub.0] in [L.sup.1] ([OMEGA]), one has similarly

[mathematical expression not reproducible]. (60)

We now pass to the limit in [mathematical expression not reproducible]. Using the hypothesis [mathematical expression not reproducible]. Since [[phi].sub.k]([u.sup.n] - [phi]) [right arrow] [[phi].sub.k](u - [phi]) weakly in [L.sup.2](0, T;[H.sup.1]([OMEGA], dv)), we have

[mathematical expression not reproducible]. (61)

Moreover,

[mathematical expression not reproducible] (62)

Since [nabla][[phi].sub.k]([u.sup.n] - [phi]) [right arrow] [nabla][[phi].sub.k] (u - [phi]) weakly in [L.sup.2](0, T; [L.sup.2]([bar.[OMEGA]], dv)), we have

[mathematical expression not reproducible]. (63)

It follows from Fatou's lemma that

[mathematical expression not reproducible]. (64)

Combining (62)-(64) we have

[mathematical expression not reproducible]. (65)

Similarly, we can obtain

[mathematical expression not reproducible]. (66)

Finally since [[phi].sub.k]([u.sup.n] - [phi]) converges to [[phi].sub.k](u - [phi]) Weakly * [L.sup.[infinity]] (0,T;[L.sup.[infinity]]([bar.[OMEGA]],dv)) and f([f.sup.n]) [right arrow] f(u) in [L.sup.1](0,T;([bar.[OMEGA]],dv)), [h.sup.n] [right arrow] h in [L.sup.-1]}([Q.sub.T]), [g.sup.n] [right arrow] g in [L.sup.1]([[GAMMA].sub.T]), we obtain that

[mathematical expression not reproducible]. (67)

Therefore passing to the limit in (57), it yields

[mathematical expression not reproducible]. (68)

Step 5. Uniqueness of entropy solutions

Assume that there is another entropy solution [bar.u] to the problem. For any T >0, taking [phi] = [u.sup.n] as a test function we have

[mathematical expression not reproducible], (69)

and taking [[phi].sub.k]([bar.u] - [u.sup.n]) as a test function in (15), we obtain that

[mathematical expression not reproducible]. (70)

Subtracting (70) from (69), it yields

[mathematical expression not reproducible]. (71)

Note that

[mathematical expression not reproducible]. (72)

Letting n [right arrow] [infinity], we have

[mathematical expression not reproducible], (73)

On the other hand, since [h.sup.n] [right arrow] h in [L.sup.1]([OMEGA]), [g.sup.n] [right arrow] g in [L.sup.1] ([GAMMA]), we obtain

[mathematical expression not reproducible]. (74)

Thanks to the assumption on f, we have

[mathematical expression not reproducible]. (75)

Since

[mathematical expression not reproducible]. (76)

Omitting them and taking n [right arrow] [infinity] in (71), we obtain

[mathematical expression not reproducible]. (77)

Then Gronwall's inequality implies that [[PSI].sub.k]([bar.u] - u)(t) = 0 for all t [member of] [0, T], and hence [bar.u] [equivalent to] u.

4. Existence of Global Attractors

This section is devoted to the proof on the existence of the global attractor for the solution semigroup [{S(i)}.sub.t [greater than or equal to] 0].

Proof of Theorem 4. We first prove continuity of the semigroup [{S(t)}.sub.t [greater than or equal to] 0] in [L.sup.1]([bar.[OMEGA]], dv). Let [{S(t)}.sub.t[greater than or equal to]0] be a family of operators corresponding to the entropy solution of problem (1) with

S(t) : ([u.sub.0], [gamma] ([u.sub.0])) [right arrow] (u (t), [gamma] (u (t))). (78)

Then for any t > 0, the map S(t) is continuous map from [L.sup.1]([bar.[OMEGA]],dv) to [L.sup.1]([bar.[OMEGA]], dv).

The proof is similar to . However, for the sack of completeness, we provide the details here. Let [u.sub.1](t) and [u.sub.2](t) be two solutions of problem (1) with initial data [u.sub.1,0], [u.sub.2,0], respectively. Assume that [u.sup.n.sub.1][(t) and [u.sup.n.sub.2](t) are the solution of the approximate problem with initial data [u.sup.n.sub.1,0], [h.sup.n.sub.1] = [h.sup.n.sub.2] and [g.sup.n.sub.1] = [g.sup.n.sub.2], respectively, such that for any T > 0 and [epsilon] > 0, there exists a no such that for any t [member of] [0, T]

[mathematical expression not reproducible]. (79)

Here we may suppose that we have already extracted from [u.sup.n.sub.i] a proper subsequence (the one converges to [u.sub.i] in C([0, T]; [L.sup.1]([bar.[OMEGA]], dv))), which is still denoted by [u.sup.n.sub.i], i = 1,2.

Note that [u.sup.n.sub.1] - [u.sup.n.sub.2] satisfies the following equation:

[mathematical expression not reproducible]. (80)

Taking (l/k)[[phi].sub.k]([u.sup.n.sub.1] - [u.sup.n.sub.2]) as a test function, and using the fact that

[mathematical expression not reproducible], (81)

we obtain that

[mathematical expression not reproducible], (82)

which implies that

[mathematical expression not reproducible]. (83)

Let k [right arrow] 0. We get

[mathematical expression not reproducible]. (84)

Therefore, for n large enough we obtain that

[mathematical expression not reproducible]. (85)

For any t > 0, [epsilon] >0 and n large enough, we know that there exists [delta] = [epsilon]/C > 0 such that

[mathematical expression not reproducible]. (86)

Since

[mathematical expression not reproducible]. (87)

In view of (79), for any [epsilon] >0 and any fixed t > 0, we can choose n large enough such that

[mathematical expression not reproducible]. (88)

Therefore, from (87) we deduce that

[mathematical expression not reproducible], (89)

which means that S(t) is continuous in [L.sup.1] ([bar.[OMEGA]], dv).

Now let us prove that [{S(t)}.sub.t[greater than or equal to]0] has an absorbing set in [L.sup.1]([bar.[OMEGA], dv); i.e., for any bounded set B [subset] [L.sup.1]([bar.[OMEGA], dv), there exist a T = T(B), such that, for all t [greater than or equal to] T, S(t)B [subset] [B.sub.0]([subset] [L.sup.1]([bar.[OMEGA],dv)). Repeating the proof of Section 3 Step 1, taking [[phi].sub.k]([u.sup.n]) as a test function in (97), we obtain that

[mathematical expression not reproducible]. (90)

Similar to the proof of (26), we get

[mathematical expression not reproducible]. (91)

Passing to the limit, we deduce that

[mathematical expression not reproducible]. (92)

Thus, the semigroup [{S(t)}.sub.t[greater than or equal to]0] possesses an absorbing set in [L.sup.1]([bar.[OMEGA]], dv), which can actually be chosen as the ball centered at zero with radius [C.sub.5]/[c.sub.1] + [[absolute value of [OMEGA]] + [absolute value of [GAMMA]] + 1 in [L.sup.1]([bar.[OMEGA], dv).

Finally, we prove that S(t) is compact in [L.sup.1]([bar.[OMEGA],dv) for any t > 0; i.e., for any bounded sequence {[u.sub.0,i], [gamma]([u.sub.o,i])}, the sequence {[u.sub.i],(t)} has convergent subsequences in [L.sup.1]([bar.[OMEGA]], dv). Here for every positive integer i, [u.sub.i](t) = S(t)[u.sub.0,i] denotes the unique entropy solution to the following problem:

[mathematical expression not reproducible]. (93)

Note, for any i, the entropy solution [u.sub.i] (t) of the above problem can be obtained by approximations as in the last section; indeed, consider the following approximate problem for (93):

[mathematical expression not reproducible], (94)

where [h.sup.n] [member of] [L.sup.[infinity]]([OMEGA]), [u.sup.n.sub.0,i] [member of] [L.sup.[infinity]]([OMEGA]), [g.sup.n] [member of] [L.sup.[infinity]]([GAMMA]) converges, respectively, to h, [u.sub.0,i], q in [L.sup.1]-norm, with

[mathematical expression not reproducible]. (95)

Thanks to the analysis in Section 3, we know that [u.sup.n.sub.i] satisfies the estimates similar to (26) and (27), i.e.,

[mathematical expression not reproducible]. (96)

Moreover, up to subsequence [u.sup.n.sub.i] converges to the unique entropy solution [u.sub.i] in C(0, T; [L.sup.1] ([bar.[OMEGA]], dv)), [[absolute value of [u.sup.n.sub.i].sup.r-1] converges to [[absolute value of [u.sub.i].sup.r-1] in [L.sup.1]([Q.sub.T],dv), and [nabla][[phi].sub.q]([u.sup.n.sub.i]) converges to [nabla][[phi].sub.k]([u.sub.i]) in [L.sup.2]([Q.sub.T], dv). Passing to the limit in n, we have

[mathematical expression not reproducible], (97)

and for any fixed T >0

[mathematical expression not reproducible], (98)

[mathematical expression not reproducible]. (99)

Here the positive constant [C.sub.10], [C.sub.11] may depend on [mathematical expression not reproducible] and [L.sup.1]-norm of the sequence {[u.sub.0,i]}, and they are independent of i. Thanks to the estimates (97)-(99), we can obtain that [mathematical expression not reproducible] is bounded in [L.sup.1](0,T;[L.sup.1]([bar.[OMEGA]],dv)) for any T > 0, and for any [q.sub.0] [member of] R satisfying 1 < [q.sub.0] < 2(r - 1)/r in [bar.[OMEGA]], we can deduce that {[u.sub.i](t)} is bounded in [mathematical expression not reproducible]. Then we deduce from the equation that

[mathematical expression not reproducible]. (100)

Thanks to the Aubin-Simon type compactness result, as is shown in Lemma 6, there exists a subsequence of {[u.sub.i]}, denoted by [mathematical expression not reproducible] which converges to a function [bar.u] in [L.sup.1](0, T; [L.sup.1]([bar.[OMEGA]], dv), i.e.,

[mathematical expression not reproducible]. (101)

Thus, up to a subsequence, [mathematical expression not reproducible] converges to [??]([tau]) in [L.sup.1]([bar.[OMEGA]], dv) for almost all [tau] [member of] (0, T). Especially for any t > 0, there exist 0 < [tau] < t such that, [mathematical expression not reproducible]. Because the map S(t) is continuous map from [L.sup.1]([bar.[OMEGA]], dv) [right arrow] [L.sup.1]([bar.[OMEGA]], dv) we can get

[mathematical expression not reproducible]. (102)

Thus, S(t) is compact in [L.sup.1]([bar.[OMEGA]], dv). From the standard results on the existence of global attractors  we conclude that the semigroup [{S(t)}.sub.t[greater than or equal to]0] possesses a global attractor A in [L.sup.1]([bar.[OMEGA]],dv), which is compact, invariant in [L.sub.1]([bar.[OMEGA]],dv) and attracts every bounded subset of [L.sup.1] ([bar.[OMEGA]], dv) in [L.sup.1]-norm.

https://doi.org/10.1155/2018/8186247

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work is supported by NSF of China (11701002) and NSF of Anhui Province (1708085MA02).

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Yonghong Duan, (1) Chunlei Hu (iD), (2) and Xiaojuan Chai (iD) (2)

(1) Department of Mathematics, Taiyuan University, Taiyuan 030032, China

(2) School of Mathematical Science, Anhui University, Hefei 230601, China

Correspondence should be addressed to Chunlei Hu; chunleihu_ahu@163.com

Received 13 March 2018; Revised 22 June 2018; Accepted 2 July 2018; Published 13 August 2018

Academic Editor: Nikos I. Karachalios
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