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

Article type: Research Article

Authors: Santos, M.L. | Rocha, M.P.C. | Ferreira, J.

Affiliations: Departamento de Matemática, Universidade Federal do Pará, Campus Universitario do Guamá, Rua Augusto Corrêa 01, Cep 66075-110, Pará, Brazil | IESAM-Instituto de Estudos Superiores da Amazônia, Av. Governador José Malcher, 1148, Belém-PA, Brazil | Departamento de Matemática, Universidade Federal de São João, São João Del Rei-MG, Brazil

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 utt+Δ2u−Δu+∫t0g1(t−s)Δu(s) ds+αut+h(u−v)=0  in \[$\widehat{Q}$, vtt+Δ2v−Δv+∫t0g2(t−s)Δv(s) ds+αvt−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.

Keywords: noncylindrical domain, strong solution, exponential decay

Journal: Asymptotic Analysis, vol. 45, no. 1-2, pp. 113-132, 2005