Global existence and new decay results of a viscoelastic wave equation with variable exponent and logarithmic nonlinearities

Abstract

In this paper, we consider the following viscoelastic problem with variable exponent and logarithmic nonlinearities: $ u_{tt}-\Delta u+u+ \int_0^tb(t-s)\Delta u(s)ds+|u_t|^{{\gamma}(\cdot)-2}u_t = u\ln{\vert u\vert^{\alpha}}, $ where $ {\gamma}(.) $ is a function satisfying some conditions. We first prove a global existence result using the well-depth method and then establish explicit and general decay results under a wide class of relaxation functions and some specific conditions on the variable exponent function. Our results extend and generalize many earlier results in the literature.

Keywords

• viscoelasticity
• relaxation function
• general decay
• logarithmic nonlinearity
• variable exponent