Global Existence and Blow-Up in a $p(x)$-Laplace Equation with Dirichlet Boundary Conditions

Abstract

This paper is devoted to a $p(x)$-Laplace equation with Dirichlet boundary. We obtain the existence of global solution to the problem by employing the method of potential wells. On the other hand, we show that the solution will blow up in finite time with $u_0 \not\equiv 0$ and nonpositive initial energy functional $J(u_0).$ By defining a positive function $F(t)$ and using the method of concavity we find an upper bound for the blow-up time.