A note on “On the existence of solutions of quasilinear wave equations with viscosity”

In the note, we study the existence and uniqueness of the global generalized solution for the following initial boundary value problem:(1)utt(t,x)−div{σ(|∇u(t,x)|2)∇u(t,x)}−Δu(t,x)−Δut(t,x)+δ|ut(t,x)|p−1ut(t,x)=μ|u(t,x)|q−1u(t,x),x∈Ω,t∈(0,T),(2)u(t,x)|∂Ω=0,t∈(0,T),(3)u(0,x)=u0(x),ut(0,x)=u1(x),x∈Ω, where Ω is a bounded domain in RN with smooth boundary ∂Ω,p≥1,q>1,δ>0,μ>0,Δ denotes the Laplacian operator in RN, ∇ denotes the gradient operator and σ(s) is a given nonlinear function.