Well-Posedness and Stability for a System of Klein-Gordon Equations

In this paper, we study the weak existence of solution for a non-linear hyperbolic coupled system of Klein-Gordon equations with memory and source terms using the Faedo-Galerkin method techniques and compactness results, we have demonstrated the uniqueness of the solution by using the classical technique. In addition, we show that the solution remains stable over time. The reaction of the proper Lyapunov function is the primary tool of the proof.

(1. 6) Several authors have studied the Klein-Gordon non-linear system among them Medeiros & M. Miranda [8] considered the non-linear system with ρ > 0 if n = 1, 2 and 1 < ρ ≤ n − 1 n − 2 if n ≥ 3, they use the argument from Komornik and Zuazua [6] to prove the existence of weak and strong solutions in Ω × (0, T ) given initial and boundary conditions. [7], considered the non-linear system with the nonlinear boundary conditions, and boundary conditions where Ω is a bounded open set of R n (n ≤ 3), α > 0 a real number, Γ 1 is a subset of the border Γ of Ω and h i a real function defined on Γ 1 × (0, ∞).
They use the Galerkin approach to demonstrate the existence of global solutions.
K. Zennir & A. Guesmia [10], considered the non-linear κth-order with non-linear sources and memory using the potential well method, they verify the existence of global solutions in the a bounded domain Ω of R n , where m i = 1, 2 are non-negative constants, r, p ≥ 2, κ ≥ 1. [5], studied the non-linear system of Klien-Gordon with acoustic boundary they demonstrate the existence of both global and weak solutions, as well as their uniqueness.
Our objective is to prove that the problem (1.1)-(1.5) has a weak and unique solution such that the kernel terms k, l have some hypothesis as well as using some ideas from articles ( [2]) and ( [9]).

Preliminaries
Let Ω be a domain in R n with smooth boundary Γ let T > 0.
The inner product and norm in L 2 (Ω) are denoted by (2.1) The norm in H 1 0 (Ω) is denoted by We assume that k, l: R + −→ R + are non-increasing differentiable functions satisfying : and If w = w (t, x) is a function in L 2 (0.T ; H 1 0 (Ω)) and k is continuous we put: Lemma 2.2.
[9] ( Young's Inequality) Let a, b ≥ 0 and 1 q + 1 p = 1 for 1 < p, q < +∞, then one has the inequality ab ≤ δa q + c(δ)b p , where δ > 0 is an arbitrary constant, and c(δ) is a positive constant depending on δ. Let s be a number with 2 ≤ s < +∞ if n ≤ 2 and 2 ≤ s ≤ 2n n−2 if n > 2. Then there is a constant C depending on Ω and s such that Then, under assumptions on two functions k and l, the problem Theorem 2.2. Let u, v :→ L 2 (Ω) be functions in the class (2.6) and (2.7) satisfying from (1.1) to

Global Existence
Step 1: Approximate solution. Using the Faedo-Galerkin process, we will determine the existence of a local solution to the problem (1.1)-(1.5) in this section. Let {w i } be a basis for both H 2 (Ω) ∩ H 1 0 (Ω) and L 2 (Ω) for each positive integer m we put we look for an approximate solution in the form Since the vectors {w i } are linearly independent, this means det(w i , w j ) = 0, the latter ensuring that Step 2: A priori estimate. Our system's energy functional E(t) is given by After that, we multiply (3.1) by u t , (3.2) by v t , and use identity (2.5) to get We found that d dt E(t) is a non-positive function, this last indicates that E(t) is a non-increasing function, meaning there exists a positive constant C 1 , independent of t and m such that From this estimation, deduce that T m = T . In addition, we get (3.7) By the Holder inequality, the embedding H 1 0 (Ω) → L 6 (Ω) and (3.7), we obtain Step 3: passage to the limit. From (3.7), (3.10), (3.11), (3.12) and (3.13) there exists a subsequence of (u m ) and a subsequence of (v m ), denoted by same symbols, such that |u m | 2 ∇v m → χ 4 weak star in L ∞ (0, T ; L 2 (Ω)). (3.14) From (3.14) and Aubin-Lions compactness Lemma in ( [3]), we obtain since ∇u m and ∇v m are bounded, then we have |v m | 2 ∇u m → |v | 2 ∇u strongly in L 2 (0, T ; L 2 (Ω)).

(3.16)
Then, there exists a subsequences of u m and v m , which we will denote by u m , v m respectively, such |v m | 2 ∇u m → |v | 2 ∇u weakly in L ∞ (0, T ; L 2 (Ω)).

(3.18)
By the last formula (3.18) and (3.14) we get Using the Cauchy Schwartz inequality, we have Since, the measure of Ω is finite, and (3.14), we obtain u m tt L 1 (Ω) ≤ C 1 .

Uniqueness
Let (u, v ) and (u 1 , v 1 ) two solutions of (1.1), we assume that U = u − u 1 and V = v − v 1 satisfy Let as put Multiplying (4.1) by U t (t) and (4.2) by (V t (t)) and summing up the product result we have From the mean value theorem, it follows that Working in the same way as in argument of Lemma (2.2) in ( [2]) there exists C > 0 such that Analogously we have and from (4.6) we have Then, by using Gronwall's lemma (1.3) in ( [4]) we get This proves the uniqueness of the solution.
Proof. We define the function of Laypunov, for > 0 as follows We prove that L(t) and E(t) are equivalent, meaning that there exist two positive constants N and M depending on such that for t ≥ 0 From the Lemma (2.2), we have By using the Poincaré inequality, we get From (3.31) we have On the other hand, we have Analogous The last term of relation (5.6) can be estimated as follow. Analogously