A characteristic of local existence for fractional heat equations in Lebesgue spaces

TL;DR

This paper characterizes the conditions on the nonlinearity for the fractional heat equation to have local solutions in Lebesgue spaces, depending on initial data and fractional order.

Contribution

It provides necessary and sufficient conditions for local existence of solutions in Lebesgue spaces for fractional heat equations with Dirichlet boundary conditions.

Findings

Solution existence depends on the growth rate of f at infinity.

Characterizations differ for q>1 and q=1 cases.

Conditions extend to the whole space when f has bounded linear growth near zero.

Abstract

In this paper, we consider the fractional heat equation $u_{t}=\triangle^{\alpha/2}u+f(u)$ with Dirichlet boundary conditions on the ball $B_{R}\subset \mathbb{R}^{d}$, where $\triangle^{\alpha/2}$ is the fractional Laplacian, $f:[0,\infty)\rightarrow [0,\infty)$ is continuous and non-decreasing. We present the characterisations of $f$ to ensure the equation has a local solution in $L^{q}(B_{R})$ provided that the non-negative initial data $u_{0}\in L^{q}(B_{R})$. For $q>1$ and $1<\alpha\leq 2$, we show that the equation has a local solution in $L^{q}(B_{R})$ if and only if $\lim_{s\rightarrow \infty}\sup s^{-(1+\alpha q/d)}f(s)=\infty$; and for $q=1$ and $1<\alpha\leq 2$ if and only if $\int_{1}^{\infty}s^{-(1+\alpha/d)}F(s)ds<\infty$, where $F(s)=\sup_{1\leq t\leq s}f(t)/t$. When $\lim_{s\rightarrow 0}f(s)/s<\infty$, the same characterisations holds for the fractional heat equation on the whole space $\mathbb{R}^{d}$.