Global Existence and Blow-Up in a $p(x)$-Laplace Equation with Dirichlet Boundary Conditions (Journal of Mathematical Study, 2019)
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 ofpotential wells. On the other hand, we show that the solution will blow up in finitetime with $u_0 ot\equiv 0$ and nonpositive initial energy functional $J(u_0).$ By defining a positivefunction $F(t)$ and using the method of concavity we find an upper bound for the blow-up time.