Some generalized results for global well-poseness for wave equations with damping and source terms

In this paper we study the initial boundary value problem of wave equations with nonlinear damping and source terms:utt−Δu+a|ut|m−1ut=b|u|p−1u,x∈Ω,t>0,u(x,0)=u0(x),ut(x,0)=u1(x),x∈Ω,u(x,t)=0,x∈∂Ω,t≥0,where Ω⊂RN is a suitably smooth bounded domain. We prove that for any a>0 and b>0, if 1<p<m<∞,u0(x)∈H01(Ω)∩Lp+1(Ω),u1(x)∈L2(Ω), then for any T>0, above problem admits a global solution u(x,t)∈L∞(0,T;H01(Ω)∩Lp+1(Ω)) with ut(x,t)∈L∞(0,T;L2(Ω))∩Lm+1(Ω×[0,T]). So the results of Georgiev and Ikehata are generalized and improved.