Global existence of solutions and energy decay for a Kirchhoff-type equation with nonlinear dissipation

This paper deals with the initial boundary value problem for a class of nonlinear Kirchhoff-type equation with dissipative term u(tt) - phi(parallel to del u parallel to(2)(2))Delta u + a vertical bar u(t)vertical bar(alpha-2)u(t) = b vertical bar u vertical bar(beta-2)u, x is an element of Omega, t > 0 in a bounded domain, where a, b > 0 and alpha, beta > 2 are constants. We obtain the global existence of solutions by constructing a stable set in H-0(1)(Omega) and show the energy decay estimate by applying a lemma of Komornik.