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: ut=−(−Δ)αu∓u2ut=−(−Δ)αu∓u2, t∈(0,T)t∈(0,T), x∈Rx∈R or TT, with 0<α≤10<α≤1 is well-posed in HsHs for s≥max⁡(−α,1/2−2α)s≥max⁡(−α,1/2−2α) except in the case α=1/2α=1/2 where it is shown to be well-posed for s>−1/2s>−1/2 and ill-posed for s=−1/2s=−1/2. As a by-product we improve the known well-posedness results for the heat equation (α=1α=1) by reaching the end-point Sobolev index s=−1s=−1. Finally, in the case 1/2<α≤11/2<α≤1, we also prove optimal results in the Besov spaces B2s,q.