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

TL;DR

This paper investigates the well-posedness of the one-dimensional quadratic fractional heat equation across different fractional orders, establishing new results especially at critical Sobolev indices and in Besov spaces.

Contribution

It provides the first comprehensive well-posedness analysis for the fractional heat equation in Sobolev and Besov spaces, including endpoint cases and improvements over previous results.

Findings

Well-posedness in Sobolev space for s ≥ max(-α, 1/2 - 2α)

Well-posedness at s > -1/2 for α=1/2, ill-posed at s=-1/2

Optimal well-posedness results in Besov spaces for 1/2<α≤1

Abstract

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