Decay of heat semigroups in \(L^\infty\) and applications to nonlinear parabolic problems in unbounded domains
type: Item
instance of: scholarly article
title: Decay of heat semigroups in \(L^\infty\) and applications to nonlinear parabolic problems in unbounded domains
zbMATH Open document ID: 0954.35032
author: Philippe Souplet
published in: Journal of Functional Analysis
publication date: 2001-02-05T00:00:00Z
review text: The author describes an interesting characterization of the Poincaré inequality on the unbounded domain \(\Omega\) in terms of the solutions \(u\) to \[ -\Delta u= f,\quad u\in L^q(\Omega),\quad f\in L^q(\Omega),\quad u=0\quad\text{on }\partial\Omega\tag{1} \] on the one hand, and in terms of the global existence of solutions \(u\) to \[ u_t=\Delta u+\lambda g(u),\quad u\in L^\infty(\Omega),\quad u=0\quad\text{on }\partial\Omega,\quad u(0)= 0,\quad\lambda> 0,\tag{2} \] on the other hand; here \(g:[0, b)\to \mathbb{R}_+\) is locally Lipschitz and satisfies \(\lim_sg(s)= \infty\) as \(s\to b\). We list two representative results; (Pi) is short for Poincaré's inequality. Theorem A: For any domain \(\Omega\subseteq \mathbb{R}^n\), (i)--(iii) below are equivalent: (i) for all \(1\leq q\leq\infty\), \(f\in L^q(\Omega)\) there is \(u\in L^q(\Omega)\) with \(-\Delta u= f\) in \({\mathcal D}'(\Omega)\), i.e. in the distributional sense, (ii) there is \(u\in L^\infty(\Omega)\) with \(-\Delta u=1\) in \({\mathcal D}'(\Omega)\), (iii) (Pi) holds on \(\Omega\). Theorem B: For any \(\Omega\subseteq \mathbb{R}^n\), (i), (ii) below are equivalent: (i) for all \(\lambda> 0\) sufficiently small, the solution \(u(t)\) to (2) exists globally, (ii) (Pi) holds on \(\Omega\). There are further interesting results of a similar kind; we refer to the paper for reasons of length. The proofs are based on subtle estimates and on a series of results on the heat semigroup, at disposal in the literature.
reviewed by: Bruno Scarpellini
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 47D06
Mathematics Subject Classification ID: 35B35
Mathematics Subject Classification ID: 35J05
Mathematics Subject Classification ID: 35K60
zbMATH DE Number: 1484765
zbMATH Keywords: Poincaré inequality
zbMATH Keywords: heat semigroup
MaRDI profile type: Publication
full work available at URL: jfan.2000.3554
OpenAlex ID: W2054834024
cites work: On kernels, eigenvalues, and eigenfunctions of operators related to elliptic problems
cites work: Decay of mass for a semilinear parabolic equation
cites work: Gaussian bounds for the Dirichlet heat kernel
cites work: The principal eigenvalue and maximum principle for second‐order elliptic operators in general domains
cites work: On the alexandroff‐bakelman‐pucci estimate and the reversed hölder inequality for solutions of elliptic and parabolic equations
cites work: Q4217277
cites work: Quenching for Semilinear Singular Parabolic Problems
cites work: A short note on quenching phenomena for semilinear parabolic equations
cites work: Q4041031
cites work: On the asymptotics of solutions of parabolic equations on unbounded domains
cites work: Stability properties of parabolic equations in unbounded domains
cites work: Q4008512
cites work: Decay of global solutions, stability and blowup for a reaction-diffusion problem with free boundary
cites work: Q4040640
cites work: Some bounds for principal frequency
cites work: On the growth of solutions of quasi‐linear parabolic equations
cites work: On solutions of initial -boundary problem for \(u_t=u_{xx}+1/(1-u)\)
cites work: Q4893314
cites work: On the lowest eigenvalue of the Laplacian for the intersection of two domains
cites work: A note on Hayman's theorem on the bass note of a drum
cites work: Geometry of unbounded domains,poincaré inequalities and stability in semilinear parabolic equations
cites work: Q3886160
cites work: On the asymptotic behaviour of solutions of nonlinear parabolic equations
DOI: 10.1006/JFAN.2000.3554
description: scientific article; zbMATH DE number 1484765