#### 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 [2â€“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 [7â€“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 .â€‰(H1) ,â€‰â€‰ and is a nondecreasing differentiable function.â€‰(H2) is a nonnegative bounded function and with â€‰(H3) The function is Lipschitz continuous and satisfies â€‰where .â€‰(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 ,â€‰â€‰, 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 PoincarĂ© 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 Greenâ€™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 HĂ¶lder 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 getâ€‰â€‰. 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*â€‰*Stepâ€‰â€‰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 .*Stepâ€‰â€‰2.* We will show that with the hypotheses (H1)â€“(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 ,â€‰â€‰, we get
Next, estimating some items of (41), by (H1), we obtain
By the HĂ¶lder inequality, and ,â€‰â€‰, we have
By assumption of the boundedness of and Sobolev embedding inequality, we get
Substituting the estimates (42)â€“(44) into (41), we obtain
Hence, there exists a subsequence of , which will be still denoted by , such thatâ€‰ weak-star in ,â€‰ weak-star in ,â€‰ 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)â€“(H3), each term of the left-hand side is nonnegative; then follows immediately.*Stepâ€‰â€‰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 ,â€‰â€‰ is a contraction mapping.*Stepâ€‰â€‰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)â€“(2) with initial data on , where depends on and is independent of . Thus, for large enough, we can get . This contradicts the maximality of . *Stepâ€‰â€‰5.* In the final step, we only need to prove the existence of the global solution. By (H3) and PoincarĂ© 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)â€“(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)â€“(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 HĂ¶lder inequality, we obtain
where . Noting that ,â€‰â€‰, 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 Youngâ€™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 Gronwallâ€™s inequality, we get