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) = -(-Delta)(alpha)u -/+ u(2), t is an element of(0,T), x is an element of R or T, with 0 < alpha <= 1 is well-posed in H-s for s >= 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 <= 1, we also prove optimal results in the Besov spaces B-2(s,q). (C) 2015 Elsevier Inc. All rights reserved.