2019/03/20 by Avy Soffer, Yifei Wu, Soffer, Avy +3 · 1 citation
Mathematics · Physics and Astronomy · #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #math-ph #math.AP #math.MP
paper · pdf · doi:10.48550/arxiv.1903.08316
16 pages. Any comments are welcome
arxiv created 2019/03/20 · arxiv updated 2019/03/21
In this paper, we consider the Cauchy global problem for the L2-critical semilinear heat equations ∂t h=Δh± |h|\frac4dh, with h(0,x)=h0, where h is an unknown real function defined on \R+×\Rd. In most of the studies on this subject, the initial data h0 belongs to Lebesgue spaces Lp(\Rd) for some p≥ 2 or to subcritical Sobolev space Hs(\Rd) with s>0. We here prove that there exists some positive constant ε0 depending on d, such that the Cauchy problem is locally and globally well-posed for any initial data h0 which is radial, supported away from origin and in the negative Sobolev space H-ε0(\Rd) including Lp(\Rd) with certain p<2 as subspace. Furthermore, unconditional uniqueness, and L2-estimate both as time t→0 and t→ +∞ were considered.