Blow-up for a nonlinear PDE with fractional Laplacian and singular quadratic nonlinearity

TL;DR

This paper investigates a fractional Laplacian PDE with a quadratic gradient nonlinearity and singular convolution term, establishing local well-posedness and conditions leading to finite-time blow-up of solutions.

Contribution

It introduces a novel analysis of blow-up phenomena for a fractional PDE with singular convolution and quadratic gradient nonlinearity, including well-posedness and blow-up criteria.

Findings

Established local existence and uniqueness of solutions.

Derived sufficient conditions for finite-time blow-up.

Identified the role of initial data and singular term in blow-up behavior.

Abstract

We consider a parabolic-type PDE with a diffusion given by a fractional Laplacian operator and with a quadratic nonlinearity of the 'gradient' of the solution, convoluted with a singular term b. 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 to ensure blow-up of the solution in finite time.