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

Published 23 Jun 2016 in math.AP | (1606.07145v1)

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 $L^{{1}_{loc}(\mathbb{R}^{n})$.}