Global existence and energy decay estimates for weak solutions to the pseudo‐parabolic equation with variable exponents

Abstract

This paper deals with the following pseudo‐parabolic equation under initial and Dirichlet boundary value conditions. The authors establish some qualitative relationships by constructing a new control function and applying the Sobolev embedding inequality, and then decay estimates are obtained due to a key integral inequality for p−>2. It is shown by the Galerkin's approximation method that weak solutions exist globally for p+ ≤ 2. Furthermore, nonextinction of the global weak solution is also obtained for negative initial energy by using a new differential inequality. These results improve and extend a recent result showed by Di, Shang, Peng (Appl. Math. Lett. 64 (2017) 67‐73).