Blow-up of solutions of a nonlinear nonlocal equation of Sobolev type

Abstract

We consider an initial-boundary value problem for the equation $$ \frac{\partial } {{\partial t}}( - \Delta ^2 u + \Delta u + \Delta _p u) + \Delta u - \left\| {\nabla u} \right\|_2^{2q} \Delta u = 0 $$ and prove a local existence theorem. By using the energy inequality method, we derive necessary and sufficient conditions for the blow-up of a solution in finite time.