2020/08/03 by Divyang G. Bhimani, Bhimani, Divyang G., Ramesh Manna +7 · 1 citation
Earth and Planetary Sciences · Mathematics · #35K05 #35S05 #42B35 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Analysis and Transform Methods #Seismic Imaging and Inversion Techniques
paper · pdf · doi:10.48550/arxiv.2008.01226
openalex publication_date 2020/08/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the Hermite operator H=-\Δ+|x|2 in \ℝd and its\nfractional powers H^\β, \β>0 in phase space. Namely, we represent\nfunctions f via the so-called short-time Fourier, alias Fourier-Wigner or\nBargmann transform Vg f (g being a fixed window function), and we measure\ntheir regularity and decay by means of mixed Lebesgue norms in phase space of\nVg f, that is in terms of membership to modulation spaces Mp,q, 0<\np,q\≤ \∞. We prove the complete range of fixed-time estimates for the\nsemigroup e-tH^\β when acting on Mp,q, for every 0< p,q\≤\n\∞, exhibiting the optimal global-in-time decay as well as phase-space\nsmoothing. As an application, we establish global well-posedness for the\nnonlinear heat equation for H\β with power-type nonlinearity (focusing\nor defocusing), with small initial data in modulation spaces or in Wiener\namalgam spaces. We show that such a global solution exhibits the same optimal\ndecay e-c t as the solution of the corresponding linear equation, where\nc=d^\β is the bottom of the spectrum of H^\β. This is in sharp\ncontrast to what happens for the nonlinear focusing heat equation without\npotential, where blow-up in finite time always occurs for (even small) constant\ninitial data - hence in M\∞,1.\n