SOLUTIONS OF KIRCHHOFF PLATE EQUATIONS WITH INTERNAL DAMPING AND LOGARITHMIC NONLINEARITY

In this article we study the existence of weak solutions for the nonlinear initial boundary value problem of the Kirchhoff equation u(tt) + Delta(2)u + M(parallel to del u parallel to(2))(-Delta u) + u(t) = u ln vertical bar u vertical bar(2), in Omega x (0, T), u(x, 0) = u(0)(x), u(t)(x, 0) = u(1)(x), x is an element of Omega, u(x, t) = partial derivative u/partial derivative eta (x, t) = 0, x is an element of partial derivative Omega, t >= 0, where Omega is a bounded domain in R-2 with smooth boundary partial derivative Omega, T > 0 is a fixed but arbitrary real number, M(s) is a continuous function on [0, +infinity) and eta is the unit outward normal on partial derivative Omega. Our results are obtained using the Galerkin method, compactness approach, potential well corresponding to the logarithmic nonlinearity, and the energy estimates due to Nakao.