International Journal of Mathematics and Mathematical Sciences
Volume 23 (2000), Issue 6, Pages 369-382
doi:10.1155/S0161171200000971

On the existence of solutions of strongly damped nonlinear wave equations

Jong Yeoul Park and Jeong Ja Bae

Department of Mathematics, Pusan National University, Pusan 609-735, Korea

Abstract

We investigate the existence and uniqueness of solutions of the following equation of hyperbolic type with a strong dissipation: utt(t,x)−(α+β(∫Ω|∇u(t,y)|2dy)γ)Δu(t,x)                                −λΔut(t,x)+μ|u(t,x)|q−1u(t,x)=0,     x∈Ω,t≥0            u(0,x)=u0(x),          ut(0,x)=u1(x),      x∈Ω,  u|∂Ω=0, where q>1,λ>0,μ∈ℝ,α,β≥0,α+β>0, and Δ is the Laplacian in ℝN.