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

In this paper, we consider the following problem utt−α△ut+△2u−△u=f(u),x∈Ω,t>0,u(x,0)=u0(x),ut(x,0)=u1(x),x∈Ω,△u(x,t)|∂Ω=u(x,t)|∂Ω=0,t≥0, where Ω⊂RnΩ⊂Rn is a bounded domain with smooth boundary. Under some assumptions of the initial data u0(x)u0(x), u1(x)u1(x) and the nonlinear function f(u)f(u), the existence of global weak solutions and global strong solutions are obtained by means of the potential well method.