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

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 $\mathbb{R}^{d}$.