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

and Applied Analysis 3 Lemma 2. For any g, u ∈ C1[0, +∞), we have

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][3][4][5][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 2 Abstract and Applied Analysis term ( ≡ 0), for the quasilinear parabolic equations with absorption term   = div (|∇| −2 ∇) −  () , (, ) ∈ Ω × (0, ∞) , (4) where Ω ⊂ R  ( ≥ 1) is a bounded domain with smooth boundary and  ≥ 2, there are many results about the global existence and finite time blow-up of solutions for the homogeneous Dirichlet boundary value problems; see [7][8][9][10][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: −2   = Δ −  (, ) , (, ) ∈ Ω × (0, ∞) , (5) where  > 1 and the strong solution tends to 0 when  → ∞ under the condition ((, ), ) > 0 but did not give the decay rate.Berrimi and Messaoudi [11] proved that if a bounded square matrix () ∈ (R + ) satisfying then the solution with small initial energy decays exponentially for  = 2 and polynomially for  > 2.
When there is a memory term ( ̸ ≡ 0), Messaoudi [12] studied the semilinear heat equation with a power form source term where the relaxation function  : R + → R + is a bounded  1 function and  > 2; 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 || −2  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: (1) that the model is representative, considering the mixed boundary value problem with a generalized Lewis function and time integral boundary conditions, and ,  are weak; (2) we give the reason and process of the definition of the energy functional; (3) 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.

Preliminaries
In the sequel we state the general hypotheses on the relaxation function , coefficient , nonlinearity , and initial value  0 .
Next, we give some important lemmas which will be used in the proof of our main results.
In order to define the energy functional () of the problem (1)-(2), we give the following computation.Multiplying (1) by   , integrating over Ω, 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 𝐸(𝑡) 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.Proof.By Poincaré inequality, we have where  is a positive constant.
Lemma 5.There exist two positive constants  1 and  2 , such that for some  > 0, we have Proof.Multiplying (1) by (), integrating over Ω, and using Green's formula, we get Differentiating (), we get Next, estimating some items of (25), combined with the definition of (), we get Combining this with (H2), (H3), ( 25), (27), and Lemma 4, we get For convenience, we take Clearly,  2 () > 0, for  > 0. We have to take appropriate  to ensure that  1 () > 0 and so we can take For some  > 0, we take positive constant  0 such that then we have This completes the proof.

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.

Proof
Step 1.We consider the following auxiliary problem for a given V: where  is the solution that we required.Giving some  > 0, we will consider the solution of the problem (34) in the space  = ([0, ]; ) ∩  1 ([0, ];  1 (Ω)) and define the norm as Step 2. We will show that with the hypotheses (H1)-(H4), for  > 0, ] ∈ , there exists a unique  ∈  which satisfies (34).
Next, we will prove the uniqueness of the solution  of (34) by contradiction argument.Let  1 ,  2 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 such that For  > 0,  > 0, we define is nonempty for taking  sufficiently large.We define a mapping  :  = (V) from   to .Firstly, we will prove that  is a contraction mapping from   to itself.From Lemma 2, we know that for any fixed V ∈   , 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.
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  max = ∞.This completes the proof.

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.
This completes the proof.
If  0 ∈  1 (Ω), then for some  > 0 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 V ∈  ∞ (0, ;  1 (Ω)) and  is a continuous function.Then there exists a positive constant , such that Then we have Proof.Applying the Hölder inequality, we obtain where which implies that If  = 1, we have Applying the above inequality and (69), we obtain This completes the proof.