On the existence of solutions of strongly damped nonlinear wave equations

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