Decay of solution for degenerate Kirchhoff equation with general nonlinearity

We consider the Kirchhoff equation u ''-Mmml:mfenced close=) open=(separators=||u||2 Delta u-Delta u '+h(u)=f in a bounded domain. We investigate the global-in-time well-posedness for a initial and boundary value problem associated to this equation, when the nonlinearity M is degenerated (M >= 0), h is a real continuous function satisfying a sign assumption and f is a given function. We also analyze the asymptotic behavior (as t ->infinity) of solutions when the antiderivative of M satisfies a growth fast than a polynomial type and h satisfies the Ambrosetti-Rabinowitz condition. In this case, a polynomial decay rate for the energy associated to problem is obtained. We also show that, in the nondegenerated case of M (M>0), the solutions decay in a exponential rate.