On inhomogeneous heat equation with inverse square potential

2022/10/18 by Divyang G. Bhimani, Bhimani, Divyang G., Saikatul Haque +1
Mathematics · #35B30 #35B40 #35K57 #35K67 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2210.09910

openalex publication_date 2022/10/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study inhomogeneous heat equation with inverse square potential, namely, ∂_tu + L_a u= ± |⋅|^-b |u|^αu, where L_a=-Δ+ a |x|^-2. We establish some fixed-time decay estimate for e^-tL_a associated with inhomogeneous nonlinearity |⋅|^-b in Lebesgue spaces. We then develop local theory in L^q- scaling critical and super-critical regime and small data global well-posedness in critical Lebegue spaces. We further study asymptotic behaviour of global solutions by using self-similar solutions, provided the initial data satisfies certain bounds. Our method of proof is inspired from the work of Slimene-Tayachi-Weissler (2017) where they considered the classical case, i.e. a=0.

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