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

Abstract

In this paper, we consider the fractional heat equation $u_t={△}^{\alpha /2}u+f(u)$ with Dirichlet boundary conditions on the ball $B_R\subset {\mathbb{R}}^d$, where ${△}^{\alpha /2}$ is the fractional Laplacian, $f:[0,\infty )\to [0,\infty )$ 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 $1<\alpha \le 2$, we show that the equation has a local solution in $L^q(B_R)$ if and only if ${\lim }_{s\to \infty }\sup s^{-(1+\alpha q/d)}f(s)=\infty$; and for $q=1$ and $1<\alpha \le 2$ if and only if ${\int }_1^{\infty }{s}^{-(1+\alpha /d)}F(s)ds<\infty$, where $F(s)={\sup }_{1\le t\le s}f(t)/t$. When ${\lim }_{s\to 0}f(s)/s<\infty$, the same characterisations holds for the fractional heat equation on the whole space ${\mathbb{R}}^d$.