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

In this paper, we consider the following viscoelastic problem with variable exponent and logarithmic nonlinearities: [u.sub.tt] - [DELTA]u + u + [[integral].sup.t.sub.0] b(t - s)[DELTA]u(s)ds + [[absolute value of [u.sub.t]].sup.[gamma](*)-2] [u.sub.t] = uln[absolute value of u] [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 Mathematics Subject Classification: 35B37; 35L55; 74D05; 93D15; 93D20