Nonlinear Functional Analysis of Boundary Value Problems 2013View this Special Issue
Global Existence and Uniform Energy Decay Rates for the Semilinear Parabolic Equation with a Memory Term and Mixed Boundary Condition
This work is concerned with a mixed boundary value problem for the semilinear parabolic equation with a memory term and generalized Lewis functions which depends on both spacial variable and time. Under suitable conditions, we prove the existence and uniqueness of global solutions and the energy functional decaying exponentially or polynomially to zero as the time goes to infinity by introducing brief Lyapunov function and precise priori estimates.
In this paper, we are concerned with the global existence and uniform energy decay rates for the nonlocal semilinear heat equation with a memory term and generalized Lewis function subjected to mixed boundary and initial conditions where is a bounded domain with sufficient smooth boundary , such that , , and , have positive measures, is the unit outward normal on , is a generalized Lewis function (when , is a positive constant, and is called Lewis function; see ) which satisfies (i)positive function and a.e. for .
Equation (1) arises naturally from a variety of mathematical models in engineering and physical science. For example, in the study of heat conduction in materials with memory, the classical Fourier's law of heat flux is replaced by the following form: where , , and the integral term represent temperature, diffusion coefficient, and the effect of memory term in the material, respectively. The study of this type of equations has drawn a considerable attention; see [2–6]. From the mathematical point of view, one would expect the integral term in the equation to be dominated by the leading term. So the theory of parabolic equations can be applied to this type of equations.
Recently, many works were dedicated to studying the global existence, blow-up solutions, and asymptotic properties of the initial boundary value problem for the parabolic equation with memory term. In the absence of the memory term , for the quasilinear parabolic equations with absorption term where is a bounded domain with smooth boundary and , there are many results about the global existence and finite time blow-up of solutions for the homogeneous Dirichlet boundary value problems; see [7–11]. The conclusions in Levine , Kalantarov, Ladyzhenskaya , and Levine et al.  showed that global and nonglobal existence depends on the nonlinearity of , , the dimension , and the initial data. For the research on global existence and asymptotic properties of the solution, we refer the readers to [10, 11]. Pucci and Serrin  studied the following equation with the homogeneous Dirichlet boundary conditions: where and the strong solution tends to 0 when under the condition but did not give the decay rate. Berrimi and Messaoudi  proved that if a bounded square matrix satisfying then the solution with small initial energy decays exponentially for and polynomially for .
When there is a memory term , Messaoudi  studied the semilinear heat equation with a power form source term where the relaxation function is a bounded function and ; he proved the existence of blow-up solution with positive initial energy and the homogeneous Dirichlet boundary condition by convexity method. Later, Fang and Sun  improved the results of  with when be replaced by fully nonlinear source term . For the study of general energy decay for the quasilinear parabolic system with a memory term, we see .
In the works mentioned above, there are few about the global existence and uniform energy decay rates of solution for parabolic equation with mixed boundary conditions. Motivated by it, we intend to study global existence and uniqueness of solutions for the mixed initial boundary value problem (1)-(2) with a memory term and generalized Lewis function by the Galerkin method and also give the estimates of uniform energy decay rates.
The main innovations of this paper are: that the model is representative, considering the mixed boundary value problem with a generalized Lewis function and time integral boundary conditions, and , are weak; we give the reason and process of the definition of the energy functional; we prove the energy decays exponentially or polynomially to zero as the time goes to infinity by introducing brief Lyapunov function and precise priori estimates.
The present work is organized as follows. In Section 2, we present the assumptions, lemmas, and energy functional for our work. In Section 3, we prove the existence and uniqueness of the global solution; Section 4 is devoted to proving the energy decay results.
In the sequel we state the general hypotheses on the relaxation function , coefficient , nonlinearity , and initial value . (H1) , and is a nondecreasing differentiable function. (H2) is a nonnegative bounded function and with (H3) The function is Lipschitz continuous and satisfies where . (H4) (Compatibility Condition) The initial value satisfies
Throughout this paper, we define that and the following scalar products and norms
To simplify the notations, we denote and by and , respectively.
Next, we give some important lemmas which will be used in the proof of our main results.
Lemma 2. For any , , we have where .
Proof. Differentiating with respect to and noting yield
This completes the proof.
We have the following properties about .
Lemma 3. The energy is nonnegative and
To show the uniform decay of the solution, we introduce a functional Here, we need to point out that denotes a positive constant not necessarily the same at different occurrences.
Lemma 4. There exists a positive constant such that
Proof. By Poincaré inequality, we have where is a positive constant.
Lemma 5. There exist two positive constants and , such that for some , we have
Proof. Multiplying (1) by , integrating over , and using Green’s formula, we get Differentiating , we get Next, estimating some items of (25), combined with the definition of , we get ; that is, By (H3), Cauchy inequality, and Hölder inequality, we have that Combining this with (H2), (H3), (25), (27), and Lemma 4, we get For convenience, we take Clearly, , for . We have to take appropriate to ensure that and . First, if , that is, , we can get . Next, if , that is, , noting that and for , we get so we can take For some , we take positive constant such that then we have , for , and This completes the proof.
3. Global Existence and Uniqueness
Step 1. We consider the following auxiliary problem for a given : where is the solution that we required. Giving some , we will consider the solution of the problem (34) in the space and define the norm as .
Step 2. We will show that with the hypotheses (H1)–(H4), for , , there exists a unique which satisfies (34).
Choose the basis in , which are orthonormal in and let be the subspace of generated by the first vectors. For any , define where satisfies the following equation: with the initial condition for any . By standard nonlinear ODE theory, we know that the problem (36) has a unique solution on some interval . The extension of the solution to the whole interval is a consequence of the first estimate, which we are going to prove below. Taking , we get that is, then, we have Integrating (40) over , , we get Next, estimating some items of (41), by (H1), we obtain By the Hölder inequality, and , , we have By assumption of the boundedness of and Sobolev embedding inequality, we get Substituting the estimates (42)–(44) into (41), we obtain Hence, there exists a subsequence of , which will be still denoted by , such that weak-star in , weak-star in , weak-star in .
Noting that , we can get . The existence of solution is proved.
Next, we will prove the uniqueness of the solution of (34) by contradiction argument. Let , be two solutions of problem (34) with the same initial values. Letting that and taking into (41), we have By (H1)–(H3), each term of the left-hand side is nonnegative; then follows immediately.
Step 3 (local existence and uniqueness). In this step, we will derive existence and uniqueness of local solution to problem (1)-(2) for appropriate small time by using contraction mapping theorem. That is, such that
For , , we define is nonempty for taking sufficiently large. We define a mapping from to .
Firstly, we will prove that is a contraction mapping from to itself. From Lemma 2, we know that for any fixed , the solution satisfies the following equation: Similar to the estimates of (42) and (43), we obtain selecting sufficiently small, then we have for taking sufficiently small, so is a mapping from to itself.
Secondly, we will prove that is a contraction mapping. Let , , where , ; then for any , we have Taking and integrating over , we get By (H3), we obtain where is located between and . Combining of (42), (55), and (54) yields that is Taking sufficiently small such that , is a contraction mapping.
Step 4. We show that if exists on , then .
We will use a standard continuation argument to prove it. Indeed, by contradiction argument, suppose that and ; then there exists a sequence and a constant , such that as and , . As we have already shown previously, for each there exists a unique solution of the problem (1)–(2) with initial data on , where depends on and is independent of . Thus, for large enough, we can get . This contradicts the maximality of .
Step 5. In the final step, we only need to prove the existence of the global solution. By (H3) and Poincaré inequality, we have It is easy to see that . This completes the proof.
4. Uniform Energy Decay Rates
In this section, we establish the estimate of uniform energy decay rates and make use of the above assumptions and preliminaries to prove the results.
Proof. Let be a positive constant. We introduce since , we get by taking large enough. From (18) and the assumption of , applying Lemma 3 and taking large enough and , we obtain that by the Gronwall's inequality which implies that Using (61), we obtain This completes the proof.
In order to prove Theorem 8, we first quote the following lemma.
Lemma 9. Assume that and is a continuous function. Then there exists a positive constant , such that Moreover, Then we have
Proof. Applying the Hölder inequality, we obtain
where . Noting that , , we obtain
which implies that
If , we have
Applying the above inequality and (69), we obtain
This completes the proof.
Proof of Theorem 8. From the assumption , we have . Integrating it over , we get Taking in Lemma 9, , then we obtain Substituting this estimate into (68), using (18) and Lemma 4, we have Applying (76), (23), and Lemma 3 and taking , we get Since , applying the Young’s inequality, from (19) and (77), we deduce that Since , that is, , then we obtain It follows from Lemma 3 that hence, we get Taking sufficiently small, using (78) and (81), we have then we obtain Let be a positive constant and define that Since , , we get by taking sufficiently large. Using (83) and Lemma 3 and taking sufficiently large, we obtain From (85) and (86), we have Applying the Gronwall’s inequality, we get