Well-posedness of the Cauchy problem for nonlinear parabolic equations with variable density in the hyperbolic space
type: Item
instance of: scholarly article
title: Well-posedness of the Cauchy problem for nonlinear parabolic equations with variable density in the hyperbolic space
zbMATH Open document ID: 1259.35004
author: Fabio Punzo
published in: NoDEA. Nonlinear Differential Equations and Applications
publication date: 2012-12-06T00:00:00Z
review text: The author studies the uniqueness and nonuniqueness of bounded solutions to nonlinear Cauchy problems with variable density of the form: \[ \begin{cases} \partial _tu=\Delta _g\left[ \Phi \left( u\right) \right] ,\,& \text{in}\,\,\mathbb{H} ^n\times \left( 0,T\right] =:\mathit{H}_T, \\ u=u_0,\,& \text{in}\,\, \mathbb{H}^n\times \left\{ 0\right\}, \end{cases} \] where \(\mathbb{H}^n\) is the \(n\)-dimensional hyperbolic space \(\left( n\geq 2\right) \), \(\Delta _g\) is the Laplace-Beltrami operator in \(\mathbb{H} ^n\), \(a\) is a positive function of the space variables and denotes the density and \(u_0\) is bounded. Under some assumptions, the author shows the following: (i) Uniqueness of bounded solutions not satisfying at infinity any additional constraints, when \(a\left( x\right) \) is bounded from below by a positive constant or it goes to infinity as \(x\) tends to infinity and \(n\geq 2\). (ii) Existence of bounded solutions satisfying at infinity some extra conditions, when \(a\left( x\right) \) goes to zero as \(x\) tends to infinity and \(n\geq 2\). The structure of the paper is the following: after an introduction (Section 1), in Section 2 the author states the uniqueness and nonuniqueness results. Section 3 is dedicated to some preliminary notions of hyperbolic geometry, while in Sections 4 and 5 the author proves the uniqueness and nonuniqueness results.
reviewed by: Cristian Chifu
Mathematics Subject Classification ID: 35A02
Mathematics Subject Classification ID: 35K59
Mathematics Subject Classification ID: 35K65
Mathematics Subject Classification ID: 35K67
Mathematics Subject Classification ID: 35A01
Mathematics Subject Classification ID: 35B51
zbMATH DE Number: 6113122
zbMATH Keywords: Laplace-Beltrami operator
zbMATH Keywords: sub- and supersolutions
MaRDI profile type: Publication
full work available at URL: s00030-011-0139-9
OpenAlex ID: W1969181987
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DOI: 10.1007/S00030-011-0139-9
description: scientific article; zbMATH DE number 6113122