On global solution of an initial boundary value problem for a class of damped nonlinear equations

Abstract In this paper, we consider the following problem u t t − α △ u t + △ 2 u − △ u = f ( u ) , x ∈ Ω , t > 0 , u ( x , 0 ) = u 0 ( x ) , u t ( x , 0 ) = u 1 ( x ) , x ∈ Ω , △ u ( x , t ) | ∂ Ω = u ( x , t ) | ∂ Ω = 0 , t ≥ 0 , where Ω ⊂ R n is a bounded domain with smooth boundary. Under some assumptions of the initial data u 0 ( x ) , u 1 ( x ) and the nonlinear function f ( u ) , the existence of global weak solutions and global strong solutions are obtained by means of the potential well method.

[1]  Glenn F. Webb,et al.  Existence and Asymptotic Behavior for a Strongly Damped Nonlinear Wave Equation , 1980, Canadian Journal of Mathematics.

[2]  Shaoyong Lai,et al.  The asymptotic solution of the Cauchy problem for a generalized Boussinesq equation , 2003 .

[3]  David H. Sattinger,et al.  On global solution of nonlinear hyperbolic equations , 1968 .

[4]  Masayoshi Tsutsumi,et al.  Existence and Nonexistence of Global Solutions for Nonlinear Parabolic Equations , 1972 .

[5]  Runzhang Xu,et al.  Fourth order wave equations with nonlinear strain and source terms , 2007 .

[6]  Lianjun An,et al.  A Weakly Nonlinear Analysis of Elasto-plastic-Microstructure Models , 1995, SIAM J. Appl. Math..

[7]  Shaoyong Lai,et al.  The global solution of an initial boundary value problem for the damped Boussinesq equation , 2004 .

[8]  Benchawan Wiwatanapataphee,et al.  On exact travelling wave solutions for two types of nonlinear K(n,n) equations and a generalized KP equation , 2008 .

[9]  J. Esquivel-Avila.  Dynamics around the ground state of a nonlinear evolution equation , 2005 .

[10]  Yang Zhijian.  Global existence, asymptotic behavior and blowup of solutions for a class of nonlinear wave equations with dissipative term☆ , 2003 .

[11]  Ryo Ikehata,et al.  Some remarks on the wave equations with nonlinear damping and source terms , 1996 .

[12]  Yang Zhijian,et al.  Existence and non‐existence of global solutions for a class of non‐linear wave equations , 2000 .

[13]  D. Sattinger,et al.  Saddle points and instability of nonlinear hyperbolic equations , 1975 .