No local $L^{1}$ solutions for semilinear fractional heat equations

TL;DR

This paper demonstrates that for certain semilinear fractional heat equations with specific initial data, no local solutions exist in the local L^1 space, highlighting limitations of solution existence under these conditions.

Contribution

It establishes the non-existence of local L^1 solutions for a class of semilinear fractional heat equations with Osgood-type nonlinearities.

Findings

Existence of initial conditions with no local L^1 solutions.

Non-existence results under Osgood-type conditions.

Highlights limitations of solution theory for fractional heat equations.

Abstract

We study the Cauchy problem for the semilinear fractional heat equation $u_{t}=\triangle^{\alpha/2}u+f(u)$ with non-negative initial value $u_{0}\in L^{q}(\mathbb{R}^{n})$ and locally Lipschitz, non-negative source term $f$. For $f$ satisfying the Osgood-type condition $\int_{1}^{\infty}\frac{ds}{f(s)}=\infty$, we show that there exist initial conditions such that the equation has no local solution in $L^{1}_{loc}(\mathbb{R}^{n})$.