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

Diego Chamorro and Elena Issoglio

Mathematische Nachrichten, 2022, vol. 295, issue 8, 1462-1479

Abstract: 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 b$\mathfrak {b}$ 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 b$\mathfrak {b}$ to ensure blow‐up of the solution in finite time.

Date: 2022
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https://doi.org/10.1002/mana.202000480

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