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

We 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 (Formula presented.) 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 (Formula presented.) to ensure blow-up of the solution in finite time.