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

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