Existence of solutions and stability results for a viscoelastic petrovsky equation with logarithmic variable-exponent nonlinearity

Abstract

In this paper, we consider a Petrovsky equation in the presence of a logarithmic nonlinearity with a variable exponent and subject to the effect of a distributed infinite memory term. In order to deal with the presence of the memory term, we adopt the history approach. First, we show that our problem is well-posed in appropriate functional spaces by means of the semigroup theory provided that the initial data are sufficiently small and the variable exponent satisfies some specific smoothness and boundedness conditions. Then, we establish explicit and general decay results for a wide class of relaxation functions having an arbitrary growth at infinity by using the multiplier method.