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 = − ( − Δ ) α u ∓ u 2 , t ∈ ( 0 , T ) , x ∈ R or T , with 0 < α ≤ 1 is well-posed in H s for s ≥ max ⁡ ( − α , 1 / 2 − 2 α ) except in the case α = 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 ( α = 1 ) by reaching the end-point Sobolev index s = − 1 . Finally, in the case 1 / 2 < α ≤ 1 , we also prove optimal results in the Besov spaces B 2 s , q .