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

We prove that the Cauchy problem associated with the one dimensional quadratic (fractional) heat equation: $u_t=D_x^{2\alpha} u \mp u^2,\; t\in (0,T),\; x\in \R$ 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 $B^{s,q}_2.$