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

This paper is concerned with the asymptotic behavior of solutions to reaction-diffusion equations with dynamic boundary conditions as well as L1-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, d]). This extends the corresponding results in the literatures.


Introduction
We consider the asymptotic behavior of solutions to the following parabolic equations with dynamic boundary conditions and irregular data, where Ω is a bounded domain in R  ( ≥ 3) with smooth boundary Γ.The first equation is the standard reactiondiffusion equation, and the second equation is the boundary equation, in which the value of  is assumed to be the trace of the function  defined for  ∈ Ω, Δ Γ  is the Laplace-Beltrami operator on Γ [1],  > 0 plays the role of a surface diffusion coefficient, and ] is the outward normal to Ω.By irregular data, we mean that  0 (), ℎ() ∈  1 (Ω), and () ∈  1 (Γ).
We also assume that  is  0 and there exist positive constants ,  1 ,  2 ,  0 > 0 and  ≥ 2 such that ( (,  1 ) −  (,  2 )) ( Partial differential equations with dynamic boundary conditions like (1) have applications in various fields such as hydrodynamics, heat transfer theory, and thermoelasticity [2][3][4][5].The existence and uniqueness of solutions for problem (1) have been studied extensively in various contexts; see, e.g., [2,[6][7][8][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  = 0, , ℎ = 0. Then in [11][12][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][18][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  2 (Ω) or even  ∞ ), and few results were known when the initial data and forcing terms are not regular, such as  1 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  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  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 [22] for elliptic equations involving measure data and was then adapted to parabolic equations with  1 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  1 (Ω, ]).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  1 or measure data have been studied extensively in the past years; see [24][25][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][28][29].The results obtained here might be viewed as an extension of the results therein to problems with more general boundary conditions.
We work within the framework of entropy solutions defined as follows.
Our first result concerns on the existence and uniqueness of entropy solutions.3) and suppose that  0 , ℎ ∈  1 (Ω) and  ∈  1 (Γ), there exists a unique entropy solution for problem (1).

Theorem 2. Under assumptions (2)-(
To give the second result on the long time behavior of solutions, we recall the definition of global attractors.Definition 3 (see [30]).Let {()} ≥0 be a semigroup on a Banach space .A subset A ⊂  is called a global attractor for the semigroup if A is compact in  and enjoy the following properties: (i) A is invariant, i.e., ()A = A for any  ≥ 0; (ii) A attracts every bounded subset of , i.e., for any bounded subset  of  and any neighborhood N(A) of the set A, there exists a  0 =  0 (, N(A)) such that Theorem 4. Assume that  0 , ℎ ∈  1 (Ω),  ∈  1 (Γ) and  satisfies assumptions ( 2)-( 3); then the semigroup generated by problem (1) admits a global attractor A in  1 (Ω, ]); i.e., A is compact, invariant in  1 (Ω, ]) and attracts every bounded subset of  1 (Ω, ]) in the norm topology of  1 (Ω, ]).
To prove the theorems above, let us first provide some preliminaries.
For  ∈ R, define the Marcinkiewicz space   (Ω, ]) as the set of measurable functions V such that for some positive constant  and all  > 0. We have the following.
The following is the well-known Aubin-Simon compactness result.

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.
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 Theorem 2 .Now provide the proof of the existence and uniqueness of entropy solutions for problem (1).For simplicity, we assume that  = 1.Let {ℎ  } ⊂  ∞ (Ω), {  } ⊂  ∞ (Γ), and   0 ⊂  ∞ (Ω) be three sequences of functions strongly convergent, respectively, to ℎ in  1 (Ω), to g in  1 (Γ) and to  0 in  1 (Ω) such that Let us consider the approximation problem of (1), By virtue of Theorem 8, there is an unique weak solution   to (15) for each , with Next, we shall follow the ideas of [24] to prove that, up to a subsequence,   converges to a measurable function , 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.
Taking   (  ) ( ≥ 1) as a test function in (15), we deduce that Since Note that  ≥ 2. From the definition of Ψ  (⋅) we obtain If we choose  = 1 and taking the above inequality in consideration, we deduce from (19) that By the standard Gronwall's inequality, we obtain that Note that By the definition of Ψ  , we have Therefore, we get for any  > 0. Furthermore, integrating (19) between 0 and , it is easy to obtain Setting  = 2/ and using (27), we deduce that Similarly we can obtain Combining ( 28) and ( 29), it yields which implies that Hence, we conclude from Lemma 5 that using Lemma 6, we know that   () is relatively compact in  1 (0, ;  1 (Ω, ])).Thus, up to a subsequence   convergence to  in  1 (0, ;  1 (Ω, ])).
Step 2. (  ) converges to () in  1 (0, ; ( Given  > 0, we define Let {   } be a sequence of real smooth increasing functions with    (0) = 0 and    () →   () as  → +∞.Taking    (  ) as a test function in (15), we deduce that for any  > 0 where Θ   is the primitive function of    .Note that the first integral is nonnegative; thus discarding it and passing to the limit in , we obtain that From ( 2), we know that when || ≥  0 we have and when || ≤  0 , we have || <  6 Combining this with (35), we obtain that for any  ∈ R where for some positive constant  3 .Thus, we deduce from (34) that (41) Now we analyze each term of the right hand side of (38); note that {ℎ  } converges to ℎ in  1 (Ω), and For any given  > 0, the first term on the right hand side can be strictly less then  whenever  >  1 .Thanks to (41), for  large enough ( >  1 ), the second term can be strictly less than  for all  (absolute continuity of the Lebesgue integral).Also, we can always find a positive constant  2 , such that Similarly, setting  4 = max{ 1 , Thus, for any  > 0, (  ) is equi-integrable in (  , ]) and due to Vitali's convergence theorem, (  ) converges to () in  1 (0, ;  1 (Ω, ])).

Existence of Global Attractors
This section is devoted to the proof on the existence of the global attractor for the solution semigroup {()} ≥0 .