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.

Exponential Growth of Solutions for a Variable-Exponent Fourth-Order Viscoelastic Equation With Nonlinear Boundary Feedback

“…and p(.) the global solutions are exponential growth (without damping term), while in the previous studies (with damping term) the authors proved that there exists a finite time such that the solutions blow up (see [4,15,17,20,24]).…”

Coupled system of nonlinear viscoelastic plate equations of (p(x),q(x))‐Kirchhoff‐type: Global existence, general decay, and blow‐up

“…Due to the great importance both theoretically and practically, numerous researchers have studied equations with nonstandard growth conditions, that is, equations with the variable exponent of nonlinearities, [31][32][33][34][35][36][37][38][39][40][41][42][43][44][45][46]. Compared with equations with variable exponent nonlinearities, less work has been done in a system of equations with variable exponent nonlinearities.…”

A Study on the Blow-Up of Solutions for a Lame´ System of Inverse Problem

“…They proved uniqueness theorem by reduction of the inverse problem to a family of equations with the M. Riesz potential. For more results on the Lamé system of inverse problems, we refer the reader to [11][12][13][14][15]25] and references therein.…”