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

Abstract

We study the Cauchy problem for the semilinear fractional heat equation $u_t={△}^{\alpha /2}u+f(u)$ 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 $L_{loc}^1({\mathbb{R}}^n)$.