General decay and blow up of solutions for a class of inverse problem with elasticity term and variable‐exponent nonlinearities

Abstract

In this paper, we study a class of Lamé inverse source problem with variable‐exponent nonlinearities. Under some suitable conditions on the coefficients and initial data, we proved general decay of solutions when the integral overdetermination tends to zero as time goes to infinity in appropriate range of variable exponents. Furthermore, in the absence of damping term, we show that there are solutions under some conditions on initial data and variable exponents which blow up in finite time.