Global existence and uniform decay of solutions for a system of wave equations with dispersive and dissipative terms

Abstract

In this paper, we consider a system of two coupled wave equations with dispersive and viscosity dissipative terms under Dirichlet boundary conditions. The global existence of weak solutions as well as uniform decay rates (exponential one) of the solution energy are established.

No Existencia Global para un Sistema de Ecuaciones de Onda Viscoelástica no Lineal

“…El resultado de la no existencia global, agrega y generaliza a los obtenidos por Liu y Yu [19], y Liu [16,17,18].…”

“…el cual fue estudiado por Liu [18] quien estableció la existencia global y el decaimiento exponencial.…”

Global Well-Posedness for a Class of Kirchhoff-Type Wave System

“…In that case, the energy increases exponentially when time goes to infinity and the initial data are large enough. Recently [31] considered a system of two coupled wave equations with dispersive and strong dissipative terms under Dirichlet boundary conditions:…”

“…By reviewing above known results and also [19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38], we will face the fact that the following unsolved problems arise naturally. Firstly, from [38] we know the global existence for the definitely positive energy, but we know less for the initial energy which may be negative.…”

Global Nonexistence for a Nonlinear Viscoelastic Equation with Nonlinear Damping and Velocity-Dependent Material Density

“…However, they do not take the global nonexistence of the solution to (4) with nonlinear source into consideration. For more information about this, readers can refer to [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18] and references therein. It is well known that the nonlinear source term | | −2 causes global nonexistence of solutions when either the condition = 0 or ℎ( ) = 0 holds in (1)(see [5][6][7][8][9][10][11][12][13][14][15] and references cited therein).…”