Solutions of Kirchhoff plate equations with internal damping and logarithmic nonlinearity

Abstract

In this article we study the existence of weak solutions for the nonlinear initial boundary value problem of the Kirchhoff equation utt + Δ2u + M(∥∇u∥2) (-Δu) + ut = u ln |u|2, in Ω x (0, T), u(x, 0) = u0(x), ut(x, 0) = u1(x), x ∈ Ω, u(x, t) = ∂u/∂η (x, t) = 0, x ∈ ∂Ω, t ≥ 0, where Ω is a bounded domain in ℝ2 with smooth boundary ∂Ω, T > 0 is a fixed but arbitrary real number, M(s) is a continuous function on [0, +∞) and η is the unit outward normal on ∂Ω. Our results are obtained using the Galerkin method, compactness approach, potential well corresponding to the logarithmic nonlinearity, and the energy estimates due to Nakao.