Uniform decay for the coupled Klein-Gordon-Schrödinger equations with locally distributed damping

Abstract

Abstract.The following coupled damped Klein-Gordon-Schrödinger equations are consideredwhere Ω is a bounded domain of R n , n ≤ 3, with smooth boundary Γ and ω is a neibourhood of ∂Ω. Here χω represents the characteristic function of ω.Assuming that a ∈ W 1,∞ (Ω) is a nonnegative function such that a(x) ≥ a0 > 0 a. e. in ω, polynomial decay rate is proved for every regular solution of the above system. Our result generalizes substantially the previous results given by the authors in the reference [CDC].

Exponential decay for the coupled Klein-Gordon-Schrödinger equations with locally distributed damping

“…Estimate for I 2 := 1 2 T 0 ω |ψ| 2 φ dx dt. Using Cauchy-Schwarz inequality, the numerical Hölder inequality, the inequality ab ≤ 1 4ε a 2 + εb 2 and taking (4), (7) and (42) into consideration, we can write…”

“…This is required in order to turn the system dissipative. Indeed, the presence of the damping terms given in (2) is not necessary by itself to guarantee that the energy E(t) associated to problem (1) (see the definition of E(t) in (7)) is a nonincreasing function of the parameter t. This will clarified in section 4. Uniform decay rate estimates to problem (1) has been considered in the previous results due to Cavalcanti et.…”

“…Uniform decay rate estimates to problem (1) has been considered in the previous results due to Cavalcanti et. al [11,7]. While in [11] a full damping was in place in both equations, in contrast, in [7] a full damping has been considered in the Schröndinger equation but just a localized damping has been considered for the wave equation.…”

“…Our result generalizes substantially the previous results given by Cavalcanti et. al in the reference [7].…”

Global Existence, Blow-up and Asymptotic Behavior of Solutions for a Class of Coupled Nonlinear Klein-Gordon Equations with Damping Terms

“…Recently, Xu [33] employed this method to study the problem (1.3) with dissipative term and obtained a sharp condition on global existence and finite blow-up of solutions. For other related results, we refer the reader to [4,6,25] and the references therein. Now, we return to the coupled Klein-Gordon equations.…”