Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Exponential energy decay and blow-up of solutions for a system of nonlinear viscoelastic wave equations with strong damping | Liang and Gao Boundary Value Problems 2011 2011 22 http content 2011 1 22 o Boundary Value Problems a SpringerOpen Journal RESEARCH Open Access Exponential energy decay and blow-up of solutions for a system of nonlinear viscoelastic wave equations with strong damping Fei Liang1 2 and Hongjun Gao1 Correspondence gaohj@hotmail. com Jiangsu Provincial Key Laboratory for Numerical Simulation of Large Scale Complex Systems School of Mathematical Sciences Nanjing Normal University Nanjing 210046 PR China Full list of author information is available at the end of the article Springer Abstract In this paper we consider the system of nonlinear viscoelastic equations utt Au f gi t T Au t dT Aut fi u v x t e n X 0 T 0 vtt Av f g2 t T Av t dT Avt f2 u v x t e n X 0 T 0 with initial and Dirichlet boundary conditions. We prove that under suitable assumptions on the functions g - f i 1 2 and certain initial data in the stable set the decay rate of the solution energy is exponential. Conversely for certain initial data in the unstable set there are solutions with positive initial energy that blow up in finite time. 2000 Mathematics Subject Classifications 35L05 35L55 35L70. Keywords decay blow-up positive initial energy viscoelastic wave equations 1. Introduction In this article we study the following system of viscoelastic equations utt Au 0tg1 t T Au t dT Aut f1 u v x t e Q X 0 T Vtt Av J0g2 t T Av t dT Avt f2 u v x t e a X 0 T u x t v x t 0 x e do. X 0 T u x 0 u0 x ut x 0 u1 x x e Q v x 0 v0 x vt x 0 v1 x x e Q where o is a bounded domain in R with a smooth boundary 90 and gj R R fi - R2 R i 1 2 are given functions to be specified later. Here u and v denote the transverse displacements of waves. This problem arises in the theory of viscoelastic and describes the interaction of two scalar fields we can refer to Cavalcanti et al. 1 Messaoudi and Tatar 2 Renardy et al. 3 . To motivate this study let us recall some results regarding single .
