Global well-posedness of wave equation with weak and strong damping terms p-Laplacian and logarithmic nonlinearity source term

Abstract

We consider the following damped p-laplacian type wave equation with logarithmic nonlinearity utt−Δut−Δpu+ut=|u|q−2uln|u|,x∈Ω,t>0 in a bounded domain with homogeneous Dirichlet boundary condition. Firstly, we prove the local existence of weak solution by using contraction mapping principle. And in the framework of potential well, we show the global existence, energy decay and when the initial energy is subcritical, a sufficient condition for the solutions to blow up in finite time is derived, by combining the Nehari manifold with concavity argument. Then we parallelly extend the conclusions of global existence and energy decay for the subcritical case to the critical case by scaling technique. Besides, When the initial energy is supercritical, some new skills are invented to establish another finite time blow-up criterion for this problem.