Global Solutions to the Initial-boundary Value Problem for the Quasilinear Viscoelastic Equation with a Derivative Nonlinearity

u(x, 0) = u0(x) and ut(x, 0) = u1(x) for x ∈ Ω, and u(x, t)|∂Ω = 0, t ≥ 0, (1.2) where Ω is a bounded domain in R with smooth, say C-class, boundary ∂Ω and σ(|∇u|) is a function like σ = 1/ √ 1 + |∇u|2, mean curvature type nonlinearity. The viscosity term −∆ut is often called a Kelvin-Voigt type dissipation or strong dissipation which appears in phenomena of wave propagation in a viscoelastic material (cf. [1, 2, 6, 14]). We make the following assumption on the nonlinear term f(u,v, w).