Decay

In this paper, we study the stability of solutions for semilinear wave equations whose boundary condition includes an integral that represents the memory eect. We show that the dissipation is strong enough to produce exponential decay of the solution, provided the relaxation function also decays exponentially. When the relaxation function decays polynomially, we show that the solution decays polynomially and with the same rate.

Frictional dissipative boundary condition for the wave equation was studied by several authors, see for example [4,5,8,9,10,11,12,16,17] among others.In these works existence of solutions and exponential stabilization were proved for linear or nonlinear equations.In contrast with the large literature for frictional dissipative, for boundary condition with memory, we have only a few works as for example [2,3,7,13,14].Let us explain briefly each of the above works.In [2] Ciarletta established theorems of existence, uniqueness and asymptotic stability for a linear model of heat conduction.In this case the memory condition describes a boundary that can absorb heat and due to the hereditary term, can retain part of it.In [3] Fabrizio & Morro consider a linear electromagnetic model with boundary condition of memory type and proved the existence, uniqueness and asymptotic stability of the solutions.While in [13] was shown the existence of global smooth solution for the one dimensional nonlinear wave equation, provided the initial data (u 0 , u 1 ) is small in the H 3 × H 2 -norm and also that the solution tends to zero as time goes to infinity.In all the above works was left open the rate of decay.In [7] Rivera & Doerty consider a nonlinear one dimensional wave equation with a viscoelastic boundary condition and proved the existence, uniqueness of global smooth solution, provided the initial data (u 0 , u 1 ) is small in the H 2 × H 1 -norm and also that the solution decays uniformly in time(exponentially or algebraically).Finally, in [14] Qin proved a blow up result for the nonlinear one dimensional wave equation with boundary condition and memory.Our main result is to show that the solution of system (1.1)-(1.3)decays uniformly in time, with rates depending on the rate of decay of the relaxation function.More precisely, denoting by k the resolvent kernel of g (the derivative of the relaxation function) we show that the solution decays exponentially to zero provided k decays exponentially to zero.When k decays polynomially, we show that the corresponding solution also decays polynomially to zero with the same rate of decay.
The method used here is based on the construction of a suitable Lyapunov functional L satisfying for some positive constants c 1 , c 2 , α and γ.To study the existence of solution of (1. The notation used in this paper is standard and can be found in Lions's book [6].In the sequel by c (sometime c 1 , c 2 , . ..) we denote various positive constants independent of t and on the initial data.The organization of this paper is as follows.In section 2 we establish a existence and regularity result.In section 3 we prove the uniform rate of exponential decay.Finally in section 4 we prove the uniform rate of polynomial decay.

. Existence and Regularity
In this section we shall study an existence and regularity of solutions to equation (1.1)- (1.3).
To this end we will assume that the relaxation function g is positive and non decreasing and we shall use (1.2) to estimate the value of µ(1, t)u x (1, t).Denoting by the convolution product operator and differentiating (1.2) we arrive at the following Volterra equation Using the Volterra inverse operator, we obtain where the resolvent kernel satisfies g .
2. Given p > 1, let us denote by c p := sup t∈R If there exists a positive constant c 0 with c 0 c p < 1 such that Proof.Note that k(0) = g(0) > 0. Now, we take 2) we get that −k * g(t 0 ) = g(t 0 ) but this is a contradiction.Therefore k(t) > 0 for all t ∈ R + 0 .Now, let us fix , such that 0 < < γ − c 0 and denote by k (t) := e t k(t), g (t) := e t g(t).

Multiplying equation (2.2) by e t we get k
Therefore which implies our first assertion.To show the second part let us consider the following notation which proves our second assertion.

Remark:
The finiteness of the constant c p can be found in [15,Lemma 7.4].
Due to this Lemma, in the remainder of this paper, we shall use (2.1) instead of (1.2).Let us denote by The following lemma states an important property of the convolution operator.
The proof of this lemma follows by differentiating the term f 2ϕ.
The first order energy of system (1.1)-(1.3) is given by We summarize the well-posedness of (1.1)-(1.3) in the following theorem. 3) then there exists only one solution u of the system (1.1)- Proof.To prove this Theorem we shall use the Galerkin method.Let {w j } be a complete orthogonal system of V such that Let us consider the following Galerkin approximation EJQTDE, 2002 No. 7, p. 5 Standard results about ordinary differential equations guarantee that there exists only one solution of the approximate system, for 0 ≤ j ≤ m, satisfying the following initial conditions Our first step is to show that the approximate solutions remain bounded for any m ∈ N. To do this, let us multiply equation (2.5) by h j,m (t) and summing up the product results in j we get 1 2 Using Lemma 2.2 for the term and the properties of k, k and k from assumption (2.3) we conclude by (2.6) Taking into account the definition of the initial data of u m we conclude that Next, we shall find an estimate for the second order energy.First, let us estimate the initial data u m tt (0) in the L 2 − norm.Letting t → 0 + in the equation (2.5), multiplying the result by h j,m (0) and using the compatibility condition (2.4) we get Since u 0 ∈ H 2 (0, 1), the hypothesis (1.6) for the function h together with the Sobolev's imbedding imply that h(u 0 ) ∈ L 2 (0, 1).Hence Differentiating the equation (2.5) with respect to the time, multiplying by h j,m (t) and summing up the product results in j, we arrive at Using the elementary inequality 2ab ≤ a 2 + b 2 , the hypothesis on µ and the inequality Substituting the inequality (2.9) and identity (2.10) into (2.8) and using Lemma 2.2 we arrive at 1 2 From the condition (1.6) and from the Sobolev imbedding we have Taking into account the first estimate (2.7) we conclude that (2.12) From the approximate system (2.5) we get Using the elementary inequality, the Sobolev's imbedding and the hypothesis (1.6) we obtain 1 2 Denoting by we find by (2.11) Substituting the inequalities (2.12) and (2.13) into (2.14) and using Poincaré's inequality , the hypothesis (2.3), the first estimate (2.7) and applying Gronwall's inequality we conclude that The rest of the proof is a matter of routine.

. Exponential Decay
In this section we shall study the asymptotic behavior of the solutions of system (1.1)-(1.3)when the resolvent kernel k is exponentially decreasing, that is, there exist positive constants b 1 , b 2 such that: The point of departure of this study is to establish some inequalities given in the next lemmas.
Let us define the functional where θ is a positive number.The following Lemma plays an important role for the construction of the Lyapunov functional.
Proof.Using the equation (1.1) we get Using the hypotheses (1.5) we have From equality (2.1), from the hypothesis for the function µ and from the inequality (3.5) we have From equality (2.1) we have , Poincare's inequality and the elementary inequality 2ab ≤ a 2 + b 2 we conclude that where c is a fixed positive constant.Substituting (3.6)-(3.7)into (3.3) and fixing = µ 0 e cτ follows the conclusion of Lemma.

Let us introduce the functional
with N > 0. It is not difficult to see that L(t) verifies where q 0 and q 1 are positive constants.We will show later that the functional L satisfies the inequality of the following Lemma.and c 0 such that then there exist positive constants γ and c such that Proof.First, let us suppose that γ 0 < γ 1 .Define F (t) by Then Integrating from 0 to t we arrive at Now, we shall assume that γ 0 ≥ γ 1 .Under these conditions we get Integrating from 0 to t we obtain Since t ≤ (γ 1 − )e (γ 1 − )t for any 0 < < γ 1 we conclude that This completes the present proof.
Finally, we shall show the main result of this section.
Theorem 3.1 Let us suppose that the initial data (u 0 , u 1 ) ∈ V × L 2 (0, 1) and that the resolvent k satisfies the conditions (3.1).Then there exist positive constants α 1 and γ 1 such that Proof.We will suppose that the initial data (u 0 , u 1 ) ∈ H 2 (0, 1) ∩ V × V and satisfies (2.3); our conclusion will follow by standard density arguments.Using the Lemmas 3.1 and 3.2 we get where q 2 > 0 is a small constant.Here we used the assumptions (3.1) in order to conclude the following estimates for the corresponding two terms appearing in the inequality (3.8).Using (3.9) we obtain Using the exponential decay of k and Lemma 3.3 we conclude for all t ≥ 0, where γ 1 = min(γ 0 , q 2 q 1 ).Use of (3.9) now completes the proof.

. Polynomial rate of decay
Here, our attention will be focused on the uniform rate of decay when the resolvent k decays polynomially as (1 + t) −p .In this case we will show that the solution also decays polynomially with the same rate.Therefore, we will assume that the resolvent kernel k satisfies for some p > 1 and some positive constants b 1 and b 2 .The following lemmas will play an important role in the sequel.
Lemma 4.1 Let m and h be integrable functions, and let 0 ≤ r < 1 and q > 0.Then, for t ≥ 0: Then using Hölder's inequality with δ = q+1 q for v and δ * = q + 1 for w we arrive at the conclusion.EJQTDE, 2002 No. 7, p. 12 Lemma 4.2 Let p > 1, 0 ≤ r < 1 and t ≥ 0. Then for r > 0, and for r = 0 Proof.The above inequalitiy is a immediate consequence of Lemma 4.1 with and t fixed.
for some positive constants c1 , c2 , α and β such that Then there exists a constant c3 > 0 such that Proof.Let us denote by Differentiating this function we have From hypothesis on f and observing that β ≥ α + 1 we get Integrating the last inequality from 0 to t, it follows This complete the present proof.Theorem 4.1 Let us suppose that the initial data (u 0 , u 1 ) ∈ V × L 2 (0, 1) and that the resolvent k satisfies the conditions (4.1).Then there is a positive constant c for which we have Proof.We will suppose that the initial data (u 0 , u 1 ) ∈ H 2 (0, 1) ∩ V × V and satisfies (2.4); our conclusion will follow by standard density arguments.We define the functional L as in (3.8) and we have the equivalence to the energy term E as give in (3.9) again.The
If g is a positive continuous function, then k also is a positive continuous function.