Remarks on the Cauchy problem for the one-dimensional quadratic (fractional) heat equation

Abstract

We prove that the Cauchy problem associated with the one dimensional quadratic (fractional) heat equation: $u_t=D_x^{2\alpha }u\mp u^2,t\in (0,T),x\in \mathbb{R}$ or $ \T $, with $0<\alpha \le 1$ is well-posed in $H^s$ for $s\ge \max (-\alpha ,1/2-2\alpha )$ except in the case $\alpha =1/2$ where it is shown to be well-posed for $s>-1/2$ and ill-posed for $s=-1/2$. As a by-product we improve the known well-posedness results for the heat equation ($\alpha =1$) by reaching the end-point Sobolev index $s=-1$. Finally, in the case $1/2<\alpha \le 1$, we also prove optimal results in the Besov spaces $B_2^{s,q}.$