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

被引:0
作者
Bui, The Anh [1 ]
Hisa, Kotaro [2 ]
机构
[1] Macquarie Univ, Dept Math & Stat, Macquarie Pk, NSW 2109, Australia
[2] Univ Tokyo, Grad Sch Math Sci, Komaba 3-8-1,Meguro Ku, Tokyo 1538914, Japan
基金
澳大利亚研究理事会;
关键词
Semilinear heat equation; Heisenberg group; Global existence; Lifespan estimates; Optimal singularities; Fractional Laplacian; GLOBAL-SOLUTIONS; FUNCTION-SPACES; HEAT-EQUATIONS; LIFE-SPAN; INEQUALITIES; NONEXISTENCE; SUPERSOLUTIONS; EXPONENT;
D O I
10.1007/s10231-024-01539-8
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
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 HN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {H}<^>N$$\end{document}. Using these conditions, we can prove that 1+2/Q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1+2/Q$$\end{document} 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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q=2N+2$$\end{document} is the homogeneous dimension of HN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {H}<^>N$$\end{document}, 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.
引用
收藏
页码:1561 / 1601
页数:41
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