Global Solutions for a System of Klein-Gordon Equations with Memory

In this paper we study the existence and uniqueness of solutions of a system of Klein-Gordon equations with memory.


Introduction
In this paper we study the global existence and uniqueness of solutions (u, v) = (u(t, x), v(t, x)) of the following nonlinear system with boundary conditions u = v = 0 in [0, T ]×∂Ω and initial conditions u(0) = u 0 , v(0) = v 0 , u t (0) = u 1 and v t (0) = v 1 in Ω.Here Ω is a bounded domain in R n , with smooth boundary, T > 0, and This system is a generalization of the following coupled system of Klein-Gordon equations 128

Doherty Andrade and Angela Mognon
where m 1 , m 2 , k 1 , k 2 are nonnegative constants, which is considered in the study of the quantum field theory.We refer the reader to Schiff [8], Segal [7] and Struwe [9] for some classical results in Klein-Gordon equations.
The generalized system (1.1)-(1.2),without memory terms, were early considered by several authors.For instance, Medeiros & Milla Miranda [2], proved the existence and uniqueness of global weak solutions.Later, da Silva Ferreira [1] proved that the first order energy decays exponentially in the presence of frictional local damping.Quite recently, Cavalcanti et al in [3] considered the asymptotic behaviour for an analogous hyperbolic-parabolic system, with boundary damping, using arguments from Komornik and Zuazua [4].
Our objective is to study the system (1.1)-(1.2) when the memory terms k * ∆u and l * ∆v have dissipative properties.More precisely, if the kernels k and l are nonnegative C 2 functions satisfying then the system has a unique strong global solution.We also use these conditions, there exist α, β > 0 such that We think that the strong solution decays uniformly as time goes to infinity.This is done by using multipliers techniques as in Muñoz Rivera [5].But because of the coupled nonlinearities f (u, v) and g(u, v), the analysis of the dissipative effect of the memory terms requires new arguments.
To simplify our analysis, we assume that Note that (1.6) holds for the classical power ρ = 2 provided that n ≤ 3.
System of Klein-Gordon Equations with Memory 129

Existence of Global Solutions
We begin with some notations that will be used throughout the paper.For the Sobolev space H 1 0 (Ω) we consider the norm u H 1 0 (Ω) = ∇u 2 , where • p denotes the standard norm in L p (Ω).The inner product in L 2 is denoted by Then, by differentiation, the following Lemma holds for w ∈ C 1 ([0, T ); Theorem 2.1 Assume that f and g satisfy condition (1.6) and k, l Assume in addition that ρ ≥ 2 and (1.4) holds.Then if ) The proof of Theorem 2.1 is based on a standard Galerkin approximation.Let {w j } be a basis for both H 1 0 (Ω) and L 2 (Ω), given by the eigenfunctions of −∆ in Ω, with Dirichlet condition.For each positive integer m we put We search for functions ) with initial conditions

Existence of Weak Solutions
It follows that E m 1 (t) is a decreasing function and hence there exists a positive constant M 1 , independent of m and t such that (2.8) From this estimate we can extend the approximate solutions (u m (t), v m (t)) to the whole interval [0, T ].In addition, we get (2.10) Therefore, going to a subsequence if necessary, there exists u, v such that (2.12) Besides, from Lions-Aubin Lemma we also have (2.13) These convergence allow us easily to pass to the limit the linear terms.
For the nonlinear terms, we get for any θ ∈ (0, ρ/(ρ − 1)), weakly in L ∞ (0, T ; L θ (Ω)).Therefore the existence of weak solutions is proved.2 To prove the existence of strong solutions we need the following two Lemmas.Lemma 2.2 Suppose that ρ ≥ 2. Then there exists a constant C > 0 independent of m and t such that (2.14) Proof.To simplify drop the upper index m and the time-variable t.
First we note that (2.15) we must assume ρ ≥ 2. But then from (1.6), we have that ρ = 2 and n ≤ 3 or ρ > 2 and n = 1, 2. Suppose ρ = 2. Then From the Sobolev imbedding H 1 0 (Ω) → L 6 (Ω) and (2.8), there exists C > 0 such that If ρ > 2 and n = 1, 2, we take System of Klein-Gordon Equations with Memory 133 since in this case H 1 0 (Ω) → L p (Ω) for all p > 1. Therefore in any case we have that (2.16) holds.Working similarly with Ω f v (u, v)v t u tt dx we conclude that The same argument shows that and the Lemma follows.2.
Lemma 2.3 There exists C > 0, depending only on the data, such that (2.17) Proof.Here we also drop the upper index m.We note that As in (2.19) we infer that Then there exists a constant Ĉ = C(k, T ) > 0 such that A similar argument proves that This ends the proof.2.
Existence of Strong Solutions: Our starting is to get second order estimates of the solutions of (1.1)-(1.2).Let us put (2.20) Then we differentiate equation (2.5) and multiply by u m tt (t) and differentiate equation (2.6) and multiply by v m tt (t).Summing up the result, we have From (2.21) and Lemma 2.2, there exists a constant C 1 > 0 such that Now we integrate the above relation from 0 to t and taking into account Lemma 2.3 and since u m tt (0) and v m tt (0) are bounded, there exists a positive constant C 2 , not depending on m, such that Then there exists a constant C 3 > 0, independently of m, such that Then from the Gronwall's Lemma we finally get a positive constant M 2 , depending on T but not on m, such that From this estimate we have and therefore Besides, from Lions-Aubin Lemma we also have Now it is a matter of routine to verify that (u, v) satisfies (2.4) and the initial conditions of the problem (1.1)-(1.2).This conclude the proof of the existence part of Theorem 2.1.Finally to prove that u ∈ L 2 (0, T ; H 2 (Ω)) for n = 3, (for n = 1 and n = 2 follows immediately from the equation).In this case ρ ≤ 2, So we have Using the equation and the resolvent operator we conclude that Then, ∆u L 2 (0,T ;L 2 ) ≤ CE 2 (0) + E 1 (0)   Working as in the proof of Lemma 2.2, there exists C > 0 such that Similarly we see that and hence from (2.30) ≤ CE 3 (t).
Then from the Gronwall's Lemma we get This proves the uniqueness statement.2