On a nonlinear coupled system for the beam equations with memory in noncylindrical domains

Abstract

We prove the exponential decay in the case n>2 as time goes to infinity of regular solutions for the nonlinear coupled system of beam equations with memory and weak damping

u_tt+Δ²u−Δu+∫^t₀g₁(t−s)Δu(s) ds+αu_t+h(u−v)=0  in $\[$\widehat{Q}$$ ,

v_tt+Δ²v−Δv+∫^t₀g₂(t−s)Δv(s) ds+αv_t−h(u−v)=0  in $\[$\widehat{Q}$$

in a noncylindrical domains of $\[$\mathbb{R}^{n+1}$$ (n≥1) under suitable hypothesis on the scalar functions g₁, g₂ and h where α is a positive constant. Observe that the coupled is nonlinear and we worked directly in the noncylindrical domain that presents some technical difficulties turning the interesting problem. We establish existence and uniqueness of regular solutions for any n≥1.