Global existence and asymptotic behaviour of solution for a damped nonlinear hyperbolic equation

In this paper we study the initial–boundary value problem for a damped nonlinear hyperbolic equation utt+Δ2u+αΔ2ut+Δf(Δu)=0,x∈Ω,t>0,u(x,0)=u0(x),ut(x,0)=u1(x),x∈Ω,u=0Δu=0,x∈∂Ω,t≥0, where Ω⊂Rn, n≥1 and α is a positive constant. Under some assumptions on f(s) and initial data, we prove the global existence of solution. Furthermore, we prove that the solution decays to zero exponentially. © 2020 Elsevier Ltd