Existence and asymptotic behaviour of solutions to weakly damped wave equations of Kirchhoff type with nonlinear damping and source terms

In this paper we consider the existence of a local solution in time to a weakly damped wave equation of Kirchhoff type with the damping term and the source term:utt(t)−M(‖tu(t)‖2)Δu(t)+γ2ut(t)+|ut(t)|put(t)=|u(t)|qu(t),x∈Ω,p>0,q>0,γ2>0 with an initial value u(0)=u0u(0)=u0, ut(0)=u1ut(0)=u1 and the Dirichlet boundary condition u(t,x)|∂Ω=0u(t,x)|∂Ω=0, where Ω   is an open bounded domain in RNRN with smooth boundary and M(s)M(s) is a locally Lipschitz function. We also discuss the global existence and exponential asymptotic behaviour of solutions.