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

In the note, we study the existence and uniqueness of the global generalized solution for the following initial boundary value problem: equation(1) View the MathML sourceutt(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), Turn MathJax on equation(2) View the MathML sourceu(t,x)|∂Ω=0,t∈(0,T), Turn MathJax on equation(3) View the MathML sourceu(0,x)=u0(x),ut(0,x)=u1(x),x∈Ω, Turn MathJax on where ΩΩ is a bounded domain in RNRN with smooth boundary View the MathML source∂Ω,p≥1,q>1,δ>0,μ>0,Δ denotes the Laplacian operator in RNRN, ∇∇ denotes the gradient operator and σ(s)σ(s) is a given nonlinear function.