Blow‐up regions for a class of fractional evolution equations with smoothed quadratic nonlinearities

Abstract

AbstractWe consider an n‐dimensional parabolic‐type PDE with a diffusion given by a fractional Laplace operator and with a quadratic nonlinearity of the “gradient” of the solution, convoluted with a term which can be singular. Our first result is the well‐posedness for this problem: We show existence and uniqueness of a (local in time) mild solution. The main result is about blow‐up of said solution, and in particular we find sufficient conditions on the initial datum and on the term to ensure blow‐up of the solution in finite time.