BLOW-UP OF SOLUTIONS OF THE INTEGRO-DIFFERENTIAL KELVIN-VOIGHT EQUATION - Shakir , А. (Bulletin of Abai KazNPU. Series of Physical and Mathematical sciences, 2022)
In this work, an initial boundary value problem for a nonlinear (but without convective term) integro-differential Kelvin-Voigt equation modified with a p-Laplacian and a nonlinear source term is investigated. The integral term in the system with a convolution is called a memory term, and it indicated the viscoelastic properties of fluids. Such system of equations is called the Oskolkov equations in some papers, and it describes the motion of incompressible viscoelastic non-Newtonian fluids. In generally, there is not unique methods to prove the existence global in time of solutions to nonlinear initial-boundary value problems. However, one can response to such type questions by establishing some qualitative properties of solutions as blow up and localization in a finite time, large time behavior, and et al. In this paper, the global in time non-existence of weak generalized solutions to the studying initial boundary value problem for nonlinear modified integro-differential Kelvin-Voigt equations is proved by establishing the blowing up in a finite time property. The blow up of weak solutions to the investigating problem is obtained by using the Kalantarov Ladyzhenskaya lemma.