Long-Time Behavior of Solutions to a Class of Parabolic Equations with Nonstandard Growth Condition

Abstract

In this paper, the author studies the long-time behavior of solutions to the p(x)-Laplace equation \(u_t=\mathrm {div}(|\nabla u|^{p(x)-2}\nabla u)+f(u)\), with homogeneous Dirichlet boundary condition in a bounded domain. As for the blow-up results, it is shown, by using the energy method, that the solutions of this problem blow up in finite time for nonpositive initial energy, or even for small positive initial energy. As for the extinction results, we give some sufficient conditions for the solutions to vanish in finite time. All these results generalize the ones when p(x) is a constant.