On Global Solution for a Class of p(x)-Laplacian Equations with Logarithmic Nonlinearity

Abstract

In this paper, we consider a class of p(x)-Laplacian equations with logarithmic source terms. Using the potential well method combined with the Nehari manifold, we prove some results on the global existence and blow-up of weak solutions in the subcritical case. Moreover, we also obtain decay estimates for the global weak solutions. Otherwise, we give an upper bound for the maximal existence time of the blow-up weak solutions under different levels of initial energy. The main difficulties here not only come from the usual difficulties in variable exponent spaces but also from the sign-changing property of the logarithmic function. Our results improve (Boudjeriou in J Elliptic Parabol Equ 6:773–794 2020) and fill the gaps in Liu et al. (Nonlinear Anal Real World Appl 64:103449, 2022). Notice that the methods here are different from Boudjeriou (2020) and can be used to extend (Zeng et al. in J Nonlinear Math Phys 29:41–57, 2022).