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

This paper deals with the following pseudo-parabolic equation u(t)-Delta u(t)-div(vertical bar del u vertical bar(m(x)-2) del u)=vertical bar u vertical bar(p(x)-2)u 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).