Article ID Journal Published Year Pages File Type
4589994 Journal of Functional Analysis 2015 23 Pages PDF
Abstract

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.

Related Topics
Physical Sciences and Engineering Mathematics Algebra and Number Theory
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