Lifespan estimates for local in time solutions to the semilinear heat equation on the Heisenberg group

Vladimir Georgiev, Alessandro Palmieri*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

3 Citations (Scopus)

Abstract

In this paper, we consider the semilinear Cauchy problem for the heat equation with power nonlinearity in the Heisenberg group Hn. The heat operator is given in this case by ∂t-ΔH, where ΔH is the so-called sub-Laplacian on Hn. We prove that the Fujita exponent 1 + 2 / Q is critical, where Q= 2 n+ 2 is the homogeneous dimension of Hn. Furthermore, we prove sharp lifespan estimates for local in time solutions in the subcritical case and in the critical case. In order to get the upper bound estimate for the lifespan (especially, in the critical case), we employ a revisited test function method developed recently by Ikeda–Sobajima. On the other hand, to find the lower bound estimate for the lifespan, we prove a local in time result in weighted L space.

Original languageEnglish
Pages (from-to)999-1032
Number of pages34
JournalAnnali di Matematica Pura ed Applicata
Volume200
Issue number3
DOIs
Publication statusPublished - 2021 Jun

Keywords

  • Critical exponent of Fujita type
  • Heisenberg group
  • Lifespan estimates
  • Semilinear heat equation
  • Test function method
  • Weighted L spaces

ASJC Scopus subject areas

  • Applied Mathematics

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