On the p-Laplacian type equation with logarithmic nonlin- earity: Existence, decay and blow up

Abstract

This work is deal with a initial boundary value problem for hyperbolic p-Laplacian type equation

u_{tt}-div(|∇u|^{p-2}∇u)-Δu_{t}=|u|^{q-2}uln|u|.

We proved, the global existence of weak solution related to the equation. Global existence results of solutions are obtained using the potential well method, Galerkin method and compactness approach corresponding to the logarithmic source term. Besides, we established the energy functional decaying polynomially to zero as the time goes to infinity due to Nakao's inequality and some precise priori estimates on logarithmic nonlinearity. For suitable conditions we proved the finite time blow up results of solutions. The proof is based on the concavity method, perturbation energy method and differential--integral inequality technique. Additionally, under suitable assumptions on initial data, the infinite time blow up result is investigated with negative initial energy.