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: u t t ( t ) − M ( ‖ t u ( t ) ‖ 2 ) Δ u ( t ) + γ 2 u t ( t ) + | u t ( t ) | p u t ( t ) = | u ( t ) | q u ( t ) , x ∈ Ω , p > 0 , q > 0 , γ 2 > 0 with an initial value u ( 0 ) = u 0 , u t ( 0 ) = u 1 and the Dirichlet boundary condition u ( t , x ) | ∂ Ω = 0 , where Ω is an open bounded domain in R N with smooth boundary and M ( s ) is a locally Lipschitz function. We also discuss the global existence and exponential asymptotic behaviour of solutions.