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

Authors: Yonghong Duan; Chunlei Hu; Xiaojuan Chai
Journal: Discrete Dynamics in Nature and Society
 (2018)

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## Abstract

This paper is concerned with the asymptotic behavior of solutions to reaction-diffusion equations with dynamic boundary conditions as well asL1-initial data and forcing terms. We first prove the existence and uniqueness of an entropy solution by smoothing approximations. Then we consider the large-time behavior of the solution. The existence of a global attractor for the solution semigroup is obtained in L1(Ω¯,dν). This extends the corresponding results in the literatures.


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## Body

## 1. Introduction

We consider the asymptotic behavior of solutions to the following parabolic equations with dynamic boundary conditions and irregular data,(1)ut-Δu+fu=hxinΩ×R+,ut-μΔΓu+∂u∂ν+fu=gxonΓ×R+,ux,0=u0inΩ,where Ω is a bounded domain in RN(N≥3) with smooth boundary Γ. The first equation is the standard reaction-diffusion 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∈Ω,  ΔΓu is the Laplace-Beltrami operator on Γ [1], μ>0 plays the role of a surface diffusion coefficient, and ν is the outward normal to Ω. By irregular data, we mean that u0(x),h(x)∈L1(Ω), and g(x)∈L1(Γ). We also assume that f is C0 and there exist positive constants l,C1,C2,c0>0 and r≥2 such that(2)C1sr-c0≤fss≤C2sr+c0∀s∈R(3)fx,s1-fx,s2s1-s2≥-ls1-s22∀s1,s2∈RPartial 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 [10] the existence of global attractors was derived under the assumption μ=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 [16], 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 toL2(Ω) or even L∞), and few results were known when the initial data and forcing terms are not regular, such as L1 functions [21]. This motivates us to investigate the existence and uniqueness results as well as the large-time behavior of solutions to problem (1) with L1 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 L1 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 [22] for elliptic equations involving measure data and was then adapted to parabolic equations with L1 data in [23]. 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 L1(Ω¯,dν). 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 andL1 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 Section2, 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 ci, C, Ci to denote generic constants in various occasions, and we will denote Ω×(0,T),  Γ×(0,T) as QT and ΓT, respectively. For a function u on QT and ΓT, we set {Ω(u≥k)}={(x,t)∈QT:u(x,t)≥k},{Γ(u≥k)}={(x,t)∈ΓT:u(x,t)≥k}.


## 2. Preliminaries and the Main Result

To deal with the dynamic boundary conditions, we introduce the Lebesgue spaces as follows (see [14] for more details). Let Ω⊂RN be a bounded domain with smooth boundary Γ. For 1≤p<1, define the Lebesgue space Lp(Ω¯,dν) as(4)LpΩ¯,dν=U=u,υ:u∈LpΩ,υ∈LpΓ,with norm(5)ULpΩ¯,dνp=∫Ωupdx+∫ΓυpdS,where dν=dx|Ω⊕dS|Γ on Ω¯ is defined by ν(A)=|A∩Ω|+S(A∩Γ) for any measurable set A⊂Ω¯. Define the Sobolev space H1(Ω¯,dν) as(6)H1Ω¯,dν=U≔u,υ:u∈H1Ω,υ∈H1Γwith norm(7)UH1Ω¯,dν2=∫Ω∇u2dx+∫Γ∇Γu2+cu2dSforc>0.It is easy to see that we can identify H1(Ω¯,dν) with H1(Ω)⊕H1(Γ) under this norm. Hereafter, we denote by H-1(Ω¯,dν) the dual space of H1(Ω¯,dν).Letφk be the truncation at height k≥0,(8)φks=ks>k,ss≤k,-ks<-k,and denote Ψk:R↦R+ as its primitive function, i.e.,(9)Ψks=∫0sφkτdτ=s22s≤k,ks-k22s≥k.It is obvious that Ψk(s)≥0 and Ψk(s)≤k|s|.We work within the framework of entropy solutions defined as follows.Definition 1.
A functionu is called an entropy solution of problem (1), if for any T>0 and k>0, u∈C(0,T;L1(Ω¯,dν))∩Lr-1(0,T;Lr-1(Ω¯;dν)); moreover,(10)∫ΩΨku-ϕTdx-∫ΩΨku-ϕ0dx+∫ΓΨku-ϕTdS-∫ΓΨku-ϕ0dS+∫QTϕtφku-ϕdxdt+∫ΓTϕtφku-ϕdSdt+∫QTfuφku-ϕdxdt+∫ΓTfuφku-ϕdSdt+∫QT∇u∇φku-ϕdxdt+μ∫ΓT∇Γu∇Γφku-ϕdSdt≤∫QThφku-ϕdxdt+∫ΓTgφku-ϕdSdt,for all k>0 and u∈C(0,T;L2(Ω¯,dν))∩L∞(0,T;L∞(Ω¯;dν)) such that ϕt∈L2(0,T;H-1(Ω¯,dν)).Our first result concerns on the existence and uniqueness of entropy solutions.Theorem 2.
Under assumptions (2)–(3) and suppose that u0,h∈L1(Ω) and g∈L1(Γ), 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 [30]).
Let{S(t)}t≥0 be a semigroup on a Banach space X. A subset A⊂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≥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 T0=T0(B,N(A)) such that(11)StB⊂NAfort≥T0.Theorem 4.
Assume thatu0,h∈L1(Ω),  g∈L1(Γ) and f satisfies assumptions (2)–(3); then the semigroup generated by problem (1) admits a global attractor A in L1(Ω¯,dν); i.e., A is compact, invariant in L1(Ω¯,dν) and attracts every bounded subset of L1(Ω¯,dν) in the norm topology of L1(Ω¯,dν).To prove the theorems above, let us first provide some preliminaries.Fors∈R, define the Marcinkiewicz space Ms(Ω¯,dν) as the set of measurable functions v such that(12)∫Ωv>kksdx+∫Γv>kksdS<C,for some positive constant C and all k>0. We have the following.Lemma 5.
Letr,s be positive constants such that r-s>0 and let u(x,t) be a function defined on (QT,dν). If u∈Mr(QT,dν), then |u|s∈L1(QT,dν). In particular, Mr(QT,dν)⊂Ls(QT,dν) for all s,r≥1 such that r-s>0.The following is the well-known Aubin-Simon compactness result.Lemma 6 (see [31]).
AssumeX⊂Z⊂Y with compact imbedding X↪Z (X,Y and Z are Banach spaces). Let F be bounded in Lp(0,T;X), where 1≤p<∞, and ∂F/∂t={∂f/∂t:f∈F} be bounded in L1(0,T;Y). Then F is relatively compact in Lp(0,T;Z).


## 3. Existence and Uniqueness of Entropy Solutions

In this section, we provide the proof for Theorem2. We begin with the existence and uniqueness results for the problem with regular data.Definition 7.
Assume thatu0,h(x)∈L2(Ω), g(x)∈L2(Γ) and f∈C0 satisfies (2)-(3). A function u is called a weak solution of problem (1), if for any T>0, and u∈C(0,T;L2(Ω¯,dν))∩Lr(0,T;Lr(Ω¯,dν))∩L2(0,T;H1(Ω¯,dν)), and moreover(13)∫Ωutφdx+∫Ω∇u∇φdx+∫ΓutφdS+μ∫Γ∇Γu∇ΓφdS+∫ΓfuφdS+∫Ωfuφdx=∫Ωhxφdx+∫Γgxφds,for all φ∈Cc∞(Ω¯).Theorem 8.
Assume thatu0,h(x)∈L2(Ω),  g(x)∈L2(Γ),  f∈C0 satisfies (2)-(3). Then problem (1) admits a weak solution u∈L2(0,T;H1(Ω¯,dν))∩C(0,T;L2(Ω¯,dν))∩Lr(0,T;Lr(Ω¯,dν)).Proof.
The proof of this theorem is based on the standard Galerkin approximation method as in [10]; we thus omit the details for concision.Proof of Theorem2.
Now provide the proof of the existence and uniqueness of entropy solutions for problem (1). For simplicity, we assume that μ=1. Let {hn}⊂L∞(Ω),  {gn}⊂L∞(Γ), and u0n⊂L∞(Ω) be three sequences of functions strongly convergent, respectively, to h in L1(Ω), to g in L1(Γ) and to u0 in L1(Ω) such that(14)u0nL1Ω≤u0L1Ω,hnL1Ω≤hL1Ω,gnL1Γ≤gL1Γ.Let us consider the approximation problem of (1),(15)utn-Δun+fun=hninΩ×R+,utn-ΔΓun+∂un∂ν+fun=gnonΓ×R+,unx,0=u0ninΩ.By virtue of Theorem 8, there is an unique weak solution un to (15) for each n, with(16)u∈C0,T;L2Ω¯,dν∩Lr0,T;LrΩ¯;dν∩L20,T;H1Ω¯;dν.Next, we shall follow the ideas of [24] to prove that, up to a subsequence, un 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 n converges to u in L1(0,T;L1(Ω¯,dν)).
Takingφk(un)(k≥1) as a test function in (15), we deduce that(17)ddt∫ΩΨkuntdx+∫ΓΨkuntdS+∫Ω∇φkun2dx+∫Γ∇Γφkun2dS+∫Ωfunφkundx+∫ΓfunφkundS≤khL1Ω+kgL1Γ.Since(18)funφkun=funsgn⁡unφkun≥C1unr-1-Cφkun,(19)ddt∫ΩΨkuntdx+∫ΓΨkuntdS+∫Ω∇φkun2dx+∫Γ∇Γφkun2dS+C1∫Ωunr-1φkundx+∫Γunr-1φkundS≤khL1Ω+kgL1Γ+CΩ+Γ.Note that r≥2. From the definition of Ψk(·) we obtain(20)0≤∫ΩΨ1undx+∫ΓΨ1undS≤∫Ωunφ1undx+∫Γunφ1undS≤∫Ωunr-1φ1undx+∫Γunr-1φ1undS+Ω+Γ.If we choose k=1 and taking the above inequality in consideration, we deduce from (19) that(21)ddt∫ΩΨ1untdx+∫ΓΨ1untdS+C1∫ΩΨ1untdx+∫ΓΨ1untdS≤hL1Ω+gl1Γ+CΩ+Γ=C5.By the standard Gronwall’s inequality, we obtain that(22)Ψ1untL1Ω¯,dν≤C5c1+e-C1tΨ1u0nL1Ω¯,dν.Note that(23)∫ΩΨ1untdx+∫ΓΨ1untdS=Ψ1untL1Ω¯,dν≤∫ΩΨ1untdx+∫ΓΨ1untdS+Ω+Γ.By the definition of Ψk, we have(24)∫ΩΨ1untdx≤untL1Ω≤∫ΩΨ1untdx+Ω,(25)∫ΓΨ1untdS≤untL1Γ≤∫ΓΨ1untdS+ΓTherefore, we get(26)untL1Ω¯,dν≤C5c1+e-C1tu0nL1Ω¯,dν+Ω+Γfor any t>0.
Furthermore, integrating (19) between 0 and T, it is easy to obtain(27)∫QT∇φkun2dxdt+∫ΓT∇Γφkun2dSdt≤C6k,∫QTunr-1dxdt+∫ΓTunr-1dSdt≤C7.Setting λ=2/r and using (27), we deduce that(28)∫Ω∇unλ>kkr-1dxdt=∫∇unλ>k∩un≤kkr-1dxdt+∫∇unλ>k∩un>kkr-1dxdt≤∫∇unλ>k∩un≤kkr-1dxdt+∫un>kkr-1dxdt≤∫un≤kkr-1∇unλk2/λdxdt+∫QTunr-1dxdt≤1k∫un≤k∇un2dxdt+∫QTunr-1dxdt≤1k∫QT∇φkun2dxdt+∫QTunr-1dxdt.Similarly we can obtain(29)∫Γ∇unλ>kkr-1dSdt≤1k∫ΓT∇Γφkun2dSdt+∫ΓTunr-1dSdt.Combining (28) and (29), it yields(30)∫Ω∇unλ>kkr-1dxdt+∫Γ∇unλ>kkr-1dSdt≤1k∫QT∇φkun2dxdt+∫ΓT∇Γφkun2dSdt+∫QTunr-1dxdt+∫ΓTunr-1dSdt≤C6+C7,which implies that |∇un|λ is bounded in Mr-1(QT,dν). Hence, we conclude from Lemma 5 that |∇un|β is bounded in L1(0,T;(Ω¯,dν)) for 1<β<2(r-1)/r, which implies that |∇un|q0 is bounded in L1(0,T;(Ω¯,dν)) for 1<q0<2(r-1)/r. Therefore, un(t) is bounded in Lq0(0,T;W1,q0(Ω¯,dν)).
Furthermore, we obtain that{utn} is bounded in Lq0(0,T;W-1,q0(Ω¯,dν))+L1(0,T;L1(Ω¯,dν)). Let X=W1,q0(Ω¯,dν),  Z=L1(Ω¯,dν),  Y=W-1,q0(Ω¯,dν)+L1(Ω¯,dν); using Lemma 6, we know that un(t) is relatively compact in L1(0,T;L1(Ω¯,dν)). Thus, up to a subsequence un convergence to u in L1(0,T;L1(Ω¯,dν)).Step 2.
f ( u n ) converges to f(u) in L1(0,T;(L1(Ω¯,dν)) for any given T>0.
By Vitali’s convergence theorem, it is enough to prove(31)limE→0⁡supn⁡∫Efundxdt+limE→0⁡supn⁡∫EfundSdt=0.Given L>0, we define(32)TLs=sgn⁡ss>L,0s≤L.Let {TLi} be a sequence of real smooth increasing functions with TLi(0)=0 and TLi(s)→TL(s) as i→+∞. Taking TLi(un) as a test function in (15), we deduce that for any T>0(33)∫ΩΘLiunTdx+∫ΓΘLiunTdS+∫QTfunTLiundxdt+∫ΓTfunTLiundSdt≤∫QThnTLiundxdt+∫ΓTgnTLiundSdt+∫ΩΘLiu0ndx+∫ΓΘLiu0ndS,where ΘLi is the primitive function of TLi. Note that the first integral is nonnegative; thus discarding it and passing to the limit in i, we obtain that(34)∫Ωun>Lfunsgn⁡undxdt+∫Γun>Lfunsgn⁡undSdt≤∫un>Lhndxdt+∫un>LgndSdt+∫Ωu0n>Lu0ndx+∫Γu0n>Lu0ndS.From (2), we know that when |s|≥σ0 we have(35)c1sr-1-Cσ0≤fssgn⁡s≤c2sr-1+Cσ0,and when |s|≤σ0, we have |f|<c6 Combining this with (35), we obtain that for any s∈R(36)c1sr-1-Cσ0-c6≤fssgn⁡s≤c2sr-1+Cσ0+c6fs≤C2sr-1+C1+c6,where C1=C/σ0,C2=max⁡{c1,c2}. This implies that(37)fs≤C3fssgn⁡s+1for some positive constant C3. Thus, we deduce from (34) that(38)∫Ωun>Lfundxdt+∫Γun>LfundSdt≤C3∫un>Lhndxdt+∫un>LgndSdt+∫Ωu0n>Lu0ndx+∫Γu0n>Lu0ndS+∫Ωu0n>Ldxdt+∫Γu0n>LdSdt.Since for any T>0, un converges to u in L10,T;Ω¯,dν; there exists a positive constant C4 independent of n, such that(39)∫Ωun>Lundxdt+∫Γun>LundSdt≤∫QTundxdt+∫ΓTundSdt<C4,for  anyL≥0.Then we have(40)Ωun>L+Γun>L=measΩun>L+measΓun>L≤1LunL1Ω¯,dν≤C4L.For any ε>0, we can always find a positive constant L1 such that(41)Ωun>L+Γun>L<ε,holds  for  alln.Now we analyze each term of the right hand side of (38); note that {hn} converges to h in L1(Ω), and(42)∫Ωun>Lhndxdt≤∫Ωun>Lhn-hdxdt+∫Ωun>Lhdxdt.For any given ε>0, the first term on the right hand side can be strictly less then ε whenever n>N1. Thanks to (41), for L large enough (L>L1), the second term can be strictly less than ε for all n (absolute continuity of the Lebesgue integral). Also, we can always find a positive constant L2, such that(43)∫Ωun>L3hndxdt<ε,forn=1,2,···N1.So setting L3=max⁡{L1,L2}), we get(44)∫Ωun>L3hndxdt<2ε,for  anyn.Similarly, setting L4=max⁡{L1,L2,L3}), we can get(45)∫Γun>L4gndSdt<2ε,for  anyn.Then for any ε>0 there exists a L5>0, such that(46)∫Ωu0n>L5u0ndx+∫Γu0n>L5u0ndS<2ε,for  anyn.Taking (41)–(46) into (38), we obtain that for any ε>0 there exists L6=max⁡{L1,L3,L4,L5}, such that(47)∫E∩Ωun>L6fundxdt+∫E∩Γun>L6fundSdt<ε,uniformly in n. On the other hand, from (36) we have(48)∫E∩Ωun>L6fundxdt+∫E∩Γun≤L6fundSdt≤∫EC2L6r-1+C2+c4<εuniformly in n, whenever |E| is small enough.
Note that for anyE⊂(QT,dν)(49)∫Efun=∫E∩Ωun>L6fundxdt+∫E∩Ωun≤L6fundSdt<ε.Thus, for any T>0, f(un) is equi-integrable in (QT,dν) and due to Vitali’s convergence theorem, f(un) converges to f(u) in L1(0,T;L1(Ω¯,dν)).Step 3.
∇ u n converges to a function ∇u in L1(0,T;L1(Ω¯,dν)).
Forε>0, taking ϕ=φε(un-um) as a test function in (15) one can deduce that(50)∫ΩΨεun-umtdx+∫ΓΨεun-umtdS+∫QT∇un-∇um2φε′un-umdxdt+∫ΓT∇Γun-∇Γum2φε′un-umdSdt≤∫QTfun-fumφεun-umdxdt+∫ΓTfun-fumφεun-umdSdt+∫QThn-hmφεun-umdxdt+∫ΓTgn-gmφεun-umdSdt+∫ΩΨεun-um0dx+∫ΓΨεun-um0dS.Note that f(un), hn, gn, and u0n are convergent in L1, and 0≤Ψε(s)≤ε, for 0<ε<1 we obtain that(51)∫QT∇un-∇um2φε′un-umdxdt+∫ΓT∇Γun-∇Γum2φε′un-umdSdt=∫QT1∇un-∇um2dxdt+∫ΓT1∇Γun-∇Γum2dSdt≤2c3ε+2c4ε+2c5ε+2c6ε<Cε,where(52)QT1=x,t∈QT:un-um≤ε,ΓT1=x,t∈ΓT:un-um≤ε.Using Hölder inequality, we get(53)∫QT∇un-∇umdxdt=∫QT1∇un-∇umdxdt+∫QT\QT1∇un-∇umdxdt≤c∫QT1∇un-∇um2dxdt1/2+Cmeasx,t:un-um>ε≤Cε1/2+CmeasA1,where A1=QT\QT1={(x,t)∈QT:|un-um|>ε}. Now let us bound meas (A1); we have(54)∫QTun-umdxdt≥∫A1un-umdxdt≥εmeasA1,un→u in L1(0,T;(Ω¯,dν)); hence, we have meas(A1)<ε. Then we have(55)∫QT∇un-∇umdxdt<ε.In the same way, we have ∫ΓT|∇Γun-∇Γum|dSdt<ε. So we can obtain that(56)∇un-∇umL10,T;L1Ω¯,dν<ε.That is {∇un} is a Cauchy sequence in L1(0,T;L1(Ω¯,dν)). Discarding the nonnegative term in the left hand side of (50) we can deduce that un→uinC(0,T;L1(Ω¯,dν)).Step 4.
Pass to the limits.
Similar to [29], taking φk(un-ϕ) as a test function in (15), we have(57)∫ΩΨkun-ϕTdx-∫ΩΨkun-ϕ0dx+∫ΓΨkun-ϕTdSdt+∫ΓΨkun-ϕ0dS+∫QTϕtφkun-ϕdxdt+∫ΓTϕtφkun-ϕdSdt+∫QTfunφkun-ϕdxdt+∫ΓTfunφkun-ϕdSdt+∫QT∇un∇φkun-ϕdxdt+∫ΓT∇Γun∇Γφkun-ϕdSdt=∫QThnφkun-ϕdxdt+∫ΓTgnφkun-ϕdSdt,where ϕ∈L2(0,T;w1,2(Ω¯,dν))∩L∞(0,T;L∞(Ω¯,dν)). Let us study the limit for n→∞ of each term.
We have seen thatun→u in C(0,T;L1(Ω¯,dν)); hence, ∀t≤T,un→uinL1(Ω¯,dν). Since Ψk is k-Lipschitz continuous one has, when n→∞(58)∫ΩΨkun-ϕTdx+∫ΓΨkun-ϕTdS→∫ΩΨku-ϕTdx+∫ΓΨku-ϕTdS.Since(59)∫ΩΨkun-ϕ0dx+∫ΓΨkun-ϕ0dS=∫ΩΨku0n-ϕ0dx+∫ΓΨku0n-ϕ0dS,and u0n→u0 in L1(Ω), one has similarly(60)∫ΩΨkun-ϕ0dx+∫ΓΨkun-ϕ0dS→∫ΩΨku0-ϕ0dx+∫ΓΨku0-ϕ0dS.We now pass to the limit in ∫QTϕtφk(un-ϕ)dxdt+∫ΓTϕtφk(un-ϕ)dSdt. Using the hypothesis ϕ∈L∞(0,T;L∞(Ω¯,dν)),  φk(un-ϕ)∈L2(0,T;H1(Ω¯,dν)). Since φk(un-ϕ)→φk(u-ϕ) weakly in L2(0,T;H1(Ω,dν)), we have(61)∫QTϕtφkun-ϕdxdt+∫ΓTϕtφkun-ϕdSdt→∫QTϕtφku-ϕdxdt+∫ΓTϕtφku-ϕdSdt.Moreover,(62)∫QT∇un∇φkun-ϕdxdt=∫QT∇un-ϕ∇φkun-ϕdxdt+∫QT∇ϕ∇φkun-ϕdxdt=∫QT∇un-ϕ2φk′un-ϕdxdt+∫QT∇ϕ∇φkun-ϕdxdt=∫QT∇φkun-ϕ2dxdt+∫QT∇ϕ∇φkun-ϕdxdt.Since ∇φk(un-ϕ)→∇φk(u-ϕ) weakly in L2(0,T;L2(Ω¯,dν)), we have(63)∫QT∇ϕ∇φkun-ϕdxdt→∫QT∇ϕ∇φku-ϕdxdt.It follows from Fatou’s lemma that(64)∫QT∇φku-ϕ2dxdt≤limn→∞⁡∫QT∇φkun-ϕ2dxdt.Combining (62)–(64) we have(65)limn→∞⁡∫QT∇un∇φkun-ϕdxdt≥∫QT1∇φku-ϕ2dxdt+∫QT1∇ϕ∇φku-ϕdxdt=∫QT1∇φku-ϕ∇φku-ϕ+∇ϕdxdt=∫QT∇u-∇ϕ+∇ϕ∇φku-ϕdxdt=∫QT∇u∇φku-ϕdxdt.Similarly, we can obtain(66)limn→∞⁡∫ΓT∇Γun∇Γφkun-ϕdSdt≥∫ΓT∇Γu∇Γφku-ϕdSdt.
Finally sinceφk(un-ϕ) converges to φk(u-ϕ)  weakly∗  L∞(0,T;L∞(Ω¯,dν)) and f(un)→f(u) in L1(0,T;(Ω¯,dν)),  hn→h in L1(QT),  gn→g in L1(ΓT), we obtain that(67)∫QTfunφkun-ϕdxdt+∫ΓTfunφkun-ϕdSdt→∫QTfuφku-ϕdxdt+∫ΓTfuφku-ϕdSdt,∫QThnφkun-ϕdxdt→∫QThφku-ϕdxdt,∫ΓTgnφkun-ϕdSdt→∫ΓTgφku-ϕdSdt.Therefore passing to the limit in (57), it yields(68)∫ΩΨku-ϕTdx-∫ΩΨku-ϕ0dx+∫ΓΨku-ϕTdS-∫ΓΨku-ϕ0dS+∫QTϕtφku-ϕdxdt+∫ΓTϕtφku-ϕdSdt+∫QTfuφku-ϕdxdt+∫ΓTfuφku-ϕdSdt+∫QT∇u∇φku-ϕdxdt+∫ΓT∇Γu∇Γφku-ϕdSdt≤∫QThφku-ϕdxdt+∫ΓTgφku-ϕdSdt.Step 5.
Uniqueness of entropy solutions
Assume that there is another entropy solutionu¯ to the problem. For any T>0, taking ϕ=un as a test function we have(69)∫ΩΨku¯-unTdx-∫ΩΨku¯-un0dx+∫ΓΨku¯-unTdS-∫ΓΨku¯-un0dS+∫QTutnφku¯-undxdt+∫ΓTutnφku¯-ϕdSdt+∫QTfu¯φku¯-undxdt+∫ΓTfu¯φku¯-undSdt+∫QT∇u¯∇φku¯-undxdt+∫ΓT∇Γu¯∇Γφku¯-undSdt≤∫QThφku¯-undxdt+∫ΓTgφku¯-undSdt,and taking φk(u¯-un) as a test function in (15), we obtain that(70)∫QTutnφku¯-undxdt+∫ΓTutnφku¯-ϕdSdt+∫QTfunφku¯-undxdt+∫ΓTfunφku¯-undSdt+∫QT∇un∇φku¯-undxdt+∫ΓT∇Γun∇Γφku¯-undSdt=∫QThnφku¯-undxdt+∫ΓTgnφku¯-undSdt.Subtracting (70) from (69), it yields(71)∫ΩΨku¯-unTdx-∫ΩΨku¯-un0dx+∫ΓΨku¯-unTdS-∫ΓΨku¯-un0dS+∫QTfu¯-unφku¯-undxdt+∫ΓTfu¯-unφku¯-undSdt+∫QT∇u¯-∇un∇φku¯-undxdt+∫ΓT∇Γu¯-∇Γun∇Γφku¯-undSdt≤∫QTh-hnφku¯-undxdt+∫ΓTg-gnφku¯-undSdt.Note that(72)Ψku¯-unT≤ku¯T+unT,Ψku¯-un0≤ku¯0+un0.Letting n→∞, we have(73)∫ΩΨku¯-unTdx+∫ΓΨku¯-unTdS→∫ΩΨku¯-uTdx+∫ΓΨku¯-uTdS,∫ΩΨku¯-un0dx+∫ΓΨku¯-un0dS→0,On the other hand, since hn→hinL1(Ω), gn→ginL1(Γ), we obtain(74)∫QTh-hnφku¯-undxdt→0,∫ΓTg-gnφku¯-undSdt→0.Thanks to the assumption on f, we have(75)-∫QTfu¯-unφku¯-undxdt≤∫QTlu¯-unφku¯-undxdt≤2l∫QTΨku¯-undxdt,-∫ΓTfu¯-unφku¯-undSdt≤∫ΓTlu¯-unφku¯-undSdt≤2l∫ΓTΨku¯-undSdt.Since(76)∫QT∇u¯-∇un∇φku¯-undxdt=∫QT∇u¯-∇un2φ′u¯-undxdt≥0,∫ΓT∇Γu¯-∇Γun∇Γφku¯-undSdt=∫ΓT∇Γu¯-∇Γun2φ′u¯-undSdt≥0.Omitting them and taking n→∞ in (71), we obtain(77)ddt∫ΩΨku¯-utdx+∫ΓΨku¯-utdS≤2l∫ΩΨku¯-utdx+∫ΓΨku¯-utdS.Then Gronwall's inequality implies that Ψk(u¯-u)(t)=0 for all t∈[0,T], and hence u¯≡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(t)}t≥0.Proof of Theorem4.
We first prove continuity of the semigroup{S(t)}t≥0 in L1(Ω¯,dν). Let {S(t)}t≥0 be a family of operators corresponding to the entropy solution of problem (1) with(78)St:u0,γu0→ut,γut.Then for any t>0, the map S(t) is continuous map from L1(Ω¯,dν) to L1(Ω¯,dν).The proof is similar to [32]. However, for the sack of completeness, we provide the details here. Let u1(t) and u2(t) be two solutions of problem (1) with initial data u1,0,u2,0, respectively. Assume that u1n(t) and u2n(t) are the solution of the approximate problem with initial data u1,0n,u2,0n, h1n=h2n and g1n=g2n, respectively, such that for any T>0 and ε>0, there exists a n0 such that for any t∈[0,T](79)uit-uintL1Ω¯,dν<ε,fori=1,2,asn≥n0.Here we may suppose that we have already extracted from uin a proper subsequence (the one converges to ui in C([0,T];L1(Ω¯,dν))), which is still denoted by uin,i=1,2.Note thatu1n-u2n satisfies the following equation:(80)u1n-u2nt-Δu1n-u2n+fu1n-fu2n=0inΩ×R+,u1n-u2nt+∂u1n-u2n∂ν-ΔΓu1n-u2n+fu1n-fu2n=0onΓ×R+,unx,0=u1,0n-u2,0ninΩ.Taking 1/kφk(u1n-u2n) as a test function, and using the fact that(81)-1k∫Ωfu1n-fu2nφku1n-u2ndx≤lk∫Ωu1n-u2nφku1n-u2ndx≤2lk∫ΩΨku1n-u2ndx,-1k∫Γfu1n-fu2nφku1n-u2ndS≤lk∫Γu1n-u2nφku1n-u2ndS≤2lk∫ΓΨku1n-u2ndS,we obtain that(82)1kddt∫ΩΨku1n-u2ndx+∫ΓΨku1n-u2ndS≤2lk∫ΩΨku1n-u2ndx+∫ΓΨku1n-u2ndS,which implies that(83)1k∫ΩΨku1n-u2ntdx+∫ΓΨku1n-u2ntdS≤e2lt1k∫ΩΨku1,0n-u2,0ndx+∫ΓΨku1,0n-u2,0ndS.Let k→0. We get(84)u1nt-u2ntL1Ω¯,dν≤e2ltu1,0n-u2,0nL1Ω¯,dν.Therefore, for n large enough we obtain that(85)u1,0n-u2,0nL1Ω¯,dν=u1,0n-u1,0+u1,0-u2,0+u2,0-u2,0nL1Ω¯,dν<Cu1,0-u2,0L1Ω¯,dν.For any t>0,ε>0 and n large enough, we know that there exists δ=ε/C>0 such that(86)u1nt-u2ntL1Ω¯,dν<εifu1,0-u2,0L1Ω¯,dν<δ.Since(87)u1t-u2tL1Ω¯,dν=u1t-u1nt+u1nt-u2nt+u2nt-u2tL1Ω¯,dν≤u1t-u1ntL1Ω¯,dν+u1nt-u2ntL1Ω¯,dν+u2nt-u2tL1Ω¯,dν.In view of (79), for any ε>0 and any fixed t>0, we can choose n large enough such that(88)uit-uintL1Ω¯,dν<ε/3,fori=1,2.Therefore, from (87) we deduce that(89)u1t-u2tL1Ω¯,dν<εas  long  asu1,0-u2,0L1Ω¯,dν<δ,which means that S(t) is continuous in L1(Ω¯,dν).Now let us prove that{S(t)}t≥0 has an absorbing set in L1(Ω¯,dν); i.e., for any bounded set B⊂L1(Ω¯,dν), there exist a T=T(B), such that, for all t≥T,  S(t)B⊂B0(⊂L1(Ω¯,dν)). Repeating the proof of Section 3 Step 1, taking φk(un) as a test function in (97), we obtain that(90)ddt∫ΩΨkuntdx+∫ΩΨkuntdS+∫Ω∇φkun2dx+∫Γ∇Γφkun2dS+∫Ωfunφkundx+∫ΓfunφkundS≤khL1Ω+kgL1Γ.Similar to the proof of (26), we get(91)untL1Ω¯,dν≤C5c1+e-C1tu0nL1Ω¯,dν+Ω+Γfor  anyt>0.Passing to the limit, we deduce that(92)utL1Ω¯,dν≤C5c1+e-C1tu0L1Ω¯,dν+Ω+Γfor  anyt>0.Thus, the semigroup {S(t)}t≥0 possesses an absorbing set in L1(Ω¯,dν), which can actually be chosen as the ball centered at zero with radius C5/c1+|Ω|+|Γ|+1 in L1(Ω¯,dν).Finally, we prove thatS(t) is compact in L1(Ω¯,dν) for any t>0; i.e., for any bounded sequence {u0,i,γ(u0,i)}, the sequence {ui(t)} has convergent subsequences in L1(Ω¯,dν). Here for every positive integer i,  ui(t)=S(t)u0,i denotes the unique entropy solution to the following problem:(93)uit-Δui+fui=hxinΩ×R+,uit-ΔΓui+∂ui∂ν+fui=gxonΓ×R+,uix,0=u0,iinΩ.Note, for any i, the entropy solution ui(t) of the above problem can be obtained by approximations as in the last section; indeed, consider the following approximate problem for (93):(94)uint-Δuin+fuin=hninΩ×R+,uint-ΔΓuin+∂uin∂ν+fuin=gnonΓ×R+,uinx,0=u0,ininΩ,where hn∈L∞(Ω),  u0,in∈L∞(Ω),  gn∈L∞(Γ) converges, respectively, to h, u0,i, g in L1-norm, with(95)u0,inL1Ω≤u0,iL1Ω,hnL1Ω≤hL1Ω,gnL1Γ≤gL1Γ.Thanks to the analysis in Section 3, we know that uin satisfies the estimates similar to (26) and (27), i.e.,(96)uintL1Ω¯,dν≤c5c1+e-c1tu0,iL1Ω¯,dν+Ω+Γfor  anyt>0,∫QT∇φkuindxdt+∫ΓT∇ΓφkuindSdt≤C6k,∫QTuinr-1dxdt+∫ΓTuinr-1dSdt≤C7.Moreover, up to subsequence uin converges to the unique entropy solution ui in C(0,T;L1(Ω¯,dν)), |uin|r-1 converges to |ui|r-1 in L1(QT,dν), and ∇φk(uin) converges to ∇φk(ui) in L2(QT,dν). Passing to the limit in n, we have(97)uitL1Ω¯,dν≤C5c1+Ω+Γ+u0,iL1Ω¯,dνe-C1tfor  anyt>0,and for any fixed T>0(98)∫QT∇φkuiτdxdt+∫ΓT∇ΓφkuiτdSdt≤C10k,(99)∫QTuiτr-1dxdt+∫ΓTuiτr-1dSdt≤C11.Here the positive constant C10,C11 may depend on ‖h‖L1(Ω),  ‖g‖L1(Γ),  |Ω|,  |Γ|,  T and L1-norm of the sequence {u0,i}, and they are independent of i. Thanks to the estimates (97)–(99), we can obtain that {|∇ui|q0} is bounded in L1(0,T;L1(Ω¯,dν)) for any T>0, and for any q0∈R satisfying 1<q0<2(r-1)/r in Ω¯, we can deduce that {ui(t)} is bounded in Lq0(0,T;W1,q0(Ω¯,dν)). Then we deduce from the equation that(100)uitis  bounded  inLq00,T;W-1,q0Ω¯,dν+L10,T;L1Ω¯,dν.Thanks to the Aubin-Simon type compactness result, as is shown in Lemma 6, there exists a subsequence of {ui}, denoted by {uik} which converges to a function u~ in L1(0,T;L1(Ω¯,dν), i.e.,(101)uik-u~L10,T;L1Ω¯,dν→0,ask→∞.Thus, up to a subsequence, uik(τ) converges to u~(τ) in L1(Ω¯,dν) for almost all τ∈(0,T). Especially for any t>0, there exist 0<τ<t such that, uikm(τ)→u~ in L1(Ω¯,dν). Because the map S(t) is continuous map from L1(Ω¯,dν)→L1(Ω¯,dν) we can get(102)uikmt=St-τuikmτ→St-τu~τinL1Ω¯,dν.Thus, S(t) is compact in L1(Ω¯,dν). From the standard results on the existence of global attractors [30] we conclude that the semigroup {S(t)}t≥0 possesses a global attractor A in L1(Ω¯,dν), which is compact, invariant in L1(Ω¯,dν) and attracts every bounded subset of L1(Ω¯,dν) in L1-norm.



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