Abstract and Applied Analysis

Volume 2013 (2013), Article ID 532935, 12 pages

http://dx.doi.org/10.1155/2013/532935

Research Article

Global Existence and Uniform Energy Decay Rates for the Semilinear Parabolic Equation with a Memory Term and Mixed Boundary Condition

Zhong Bo Fang and Liru Qiu

School of Mathematical Sciences, Ocean University of China, Qingdao 266100, China

Received 14 July 2013; Accepted 18 September 2013

Academic Editor: Shaoyong Lai

Copyright © 2013 Zhong Bo Fang and Liru Qiu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

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.

1. Introduction

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 [1]) 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 [7], Kalantarov, Ladyzhenskaya [8], and Levine et al. [9] 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 [10] 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 [11] 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 [12] 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 [13] improved the results of [12] 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 [14].

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.

2. Preliminaries

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

Remark 1. The condition is necessary to guarantee the parabolicity of the problem (1)-(2).

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

which implies

This completes the proof.

In order to define the energy functional of the problem (1)-(2), we give the following computation. Multiplying (1) by , integrating over , and using Green's formula, we get from Lemma 2 that

where .

The above computation inspires us to define the energy functional of the problem (1)-(2) as

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

In this section, we show the existence and uniqueness of the global solution to problem (1)-(2) by the Galerkin method, contraction mapping principle, and contradiction argument.

Theorem 6. Assume that (H1)-(H4) holds; there exists a unique global solution of the problem (1)-(2).

Proof 

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.

Theorem 7. Assume that (H1)–(H4) hold and there exists a positive constant such that . If , then for some , there exists a positive constant such that the solution of (1)-(2) satisfies

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.

Theorem 8. Assume that (H1)–(H4) hold, and there exists a positive constant such that , . If , then for some there exists a positive constant , such that the solution of (1)-(2) satisfies

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

hence

Since , we have

and then we obtain

Using Lemma 9, we get

By (92) and replacing the left-hand side term of (78) by , we deduce that

This completes the proof.

Conflict of Interests

The authors declare that they have no conflict of interests.

Authors’ Contribution

All authors contributed equally to the paper and read and approved the final paper.

Acknowledgments

This work is supported by the Natural Science Foundation of Shandong Province of China (ZR2012AM018) and the Fundamental Research Funds for the Central Universities (no. 201362032). The authors would like to deeply thank all the reviewers for their insightful and constructive comments.

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