Global existence and energy decay estimate of solutions for a higher-order Kirchhoff type equation with damping and source term

This paper deals with the initial–boundary value problem for a class of nonlinear higher-order Kirchhoff-type equation with dissipative term utt+‖A12u‖2pAu+a|ut|q−2ut=b|u|r−2u,x∈Ω,t>0 in a bounded domain, where A=(−Δ)mA=(−Δ)m, m>1m>1 is a positive integer, a,b,p>0a,b,p>0 and q,r>2q,r>2 are constants. We obtain the global existence of solutions by constructing a stable set in H0m(Ω) and show the energy decay estimate by applying a lemma of V. Komornik.