Global existence and blow-up of image solutions for a class of nonlinear wave equations with dispersive term Original Research Article

In this paper, we study the initial boundary value problem View the MathML sourceutt-αuxxt-uxxtt=σ(ux)x,(α⩾0),x∈Ω,t>0,u(x,0)=u0(x),ut(x,0)=u1(x),x∈Ω,u(0,t)=u(1,t)=0,t⩾0, Turn MathJax on where Ω=(0,1)Ω=(0,1). We first prove that if σ(s)∈Ck(R)σ(s)∈Ck(R)(k⩾2)(k⩾2) and satisfies (H1)(H1) σ(s)σ(s) is bounded below, i.e. σ′(s)⩾C0σ′(s)⩾C0 for some constant C0C0 (H2)(H2) View the MathML source|σ1(s)|⩽C1∫0sσ1(τ)dτ+C2, where σ1(s)=σ(s)-K0s-σ(0)σ1(s)=σ(s)-K0s-σ(0), K0=min{C0,0}K0=min{C0,0}, View the MathML sourceu0(x),u1(x)∈Wk,p(Ω)∩W01,p(Ω),10T>0 the problem admits a unique solution View the MathML sourceu(x,t)∈W2,∞(0,T;Wk,p(Ω)∩W01,p(Ω)). Then the asymptotic behaviors and blow-up of solutions are discussed in detail.