AN INITIAL-BOUNDARY VALUE PROBLEM FOR KELVIN-VOIGT EQUATIONS WITH (p(x), q(x),m(x))- STRUCTURE

Abstract

The proof of the global in time existence of solutions of initial-boundary value problems for nonlinear equations in many The proof of the global in time existence of solutions of initial-boundary value problems for nonlinear equations in many cases,  is not easy, even in some cases it is impossible. However, by showing some qualitative properties of its solution, one can find answers to such questions. For example, by establishing the blow up in a finite time property of a solution, one can show that a solution does not exist globally in time. Such way, in last years, the investigating the quality properties of solutions as localization and/or blow up in a finite time, has been developing rapidly In this paper, we study the initial-boundary value problem for the Kelvin-Voigt equations with both of diffusion and relaxation term, modified by (p(x),q(x))-Laplacian, respectively, and with the variable exponent damping term. The damping term in the equation realizes as nonlinear source term. In this work, we study the nonlinear initial-boundary value problem for generalized Kelvin-Voigt equations describing the motion of incompressible viscoelastic non-Newtonian fluids. The equations generalized by replacing the diffusion and relaxation terms in equation with p(x)-Laplacian and q(x)-Laplacian, respectively, and adding a nonlinear absorption term with variable exponents and coefficients. The definition of a weak solution is given. Under suitable conditions for variable exponents and coefficients, and data of the problem, the blow up of weak solutions are established.