Existence of solutions semilinear parabolic equations with singular initial data in the Heisenberg group

2024/08/30 by The Anh Bui, Bui, The Anh, Kotaro Hisa +1
Engineering · Mathematics · #35A01 #35K15 #35R03 #35R11 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Stability and Controllability of Differential Equations #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2408.16985

openalex publication_date 2024/08/30 · openalex created_date 2024/10/19 · openalex updated_date 2026/07/28

Abstract

In this paper we obtain necessary conditions and sufficient conditions on the initial data for the solvability of fractional semilinear heat equations with power nonlinearities in the Heisenberg group ℍ^N. Using these conditions, we can prove that 1+2/Q separates the ranges of exponents of nonlinearities for the global-in-time solvability of the Cauchy problem (so-called the Fujita-exponent), where Q=2N+2 is the homogeneous dimension of ℍ^N, and identify the optimal strength of the singularity of the initial data for the local-in-time solvability. Furthermore, our conditions lead sharp estimates of the life span of solutions with nonnegative initial data having a polynomial decay at the space infinity.

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