Blow-up and Decay for a Pseudo-parabolic Equation with Nonstandard Growth Conditions

Abstract

This paper deals with a pseudo-parabolic equation involving variable exponents under homogeneous Dirichlet boundary value condition. The authors first develop the potential well method to prove a threshold result on the existence or nonexistence of global solutions to the equation when the initial energy is less than the mountain pass level $d$. In the case of high energy initial data, a new characterization for the nonexistence of global solution is also given. These results extend and improve some recent results obtained by Di et al. [9].