Well‐Posedness and Blow‐Up Results for a <i>p</i> ( <i>x</i> )‐Laplacian Viscoelastic Wave Equation With Logarithmic Source Term

ABSTRACT This paper investigates a class of ‐Laplacian‐viscoelastic wave equations with logarithmic source terms, defined on a bounded domain . First, we prove the local existence of solutions by using the Faedo–Galerkin method. Next, by combining analytical techniques, including the potential well method and energy estimates, we establish the global existence of solutions. Furthermore, employing the Komornik inequality, we prove the asymptotic stability of these solutions. Finally, through an integration of the potential well framework with concavity arguments, we demonstrate the occurrence of blow‐up phenomena under conditions of positive initial energy and derive an upper bound for the blow‐up time.