Triviality of bounded solutions and gradient estimates for nonlinear \(f\)-heat equations on complete smooth metric measure spaces
type: Item
instance of: scholarly article
title: Triviality of bounded solutions and gradient estimates for nonlinear \(f\)-heat equations on complete smooth metric measure spaces
author: Abimbola Abolarinwa
author: Akram Ali
author: Fatemah Y. Y. Mofarreh
published in: Journal of Geometry and Physics
publication date: 2022-10-18T00:00:00Z
review text: The authors analyze qualitative properties of positive smooth solutions to a generalized nonlinear parabolic equation equation involving the \(f\)-Laplacian \(\mathcal{L}_f\), given by \[ \left(\mathcal{L}_f-q(x,t)-\frac{\partial}{\partial t}\right)w(x,t) = \mathcal{G}(w(x,t)) , \] on complete smooth metric measure spaces \(M_f\times (-\infty,\infty)\), with or without boundary. They assume that the potential function \(q(x,t)\) is at least \(C^1\) in \(x\) and at least \(C^0\) in \(t\), and that the nonlinear sourcing term \(\mathcal{G}(w(x,t))\) is at least \(C^1\) in \(w>0\). The article is divided into two parts. In the first part, the authors consider spaces without boundary and establish local and global type space only (elliptic type) gradient estimates for the equation under the condition that the Bakry-Émery Ricci curvature tensor is bounded from below. They then obtain a parabolic Harnack inequality and some Liouville-type theorems for bounded ancient and eternal solutions using these gradient estimates. In the second part, the authors obtain similar results for weighted manifolds with compact boundary by imposing a Dirichlet boundary condition and assuming lower bounds on the weighted mean curvature of the boundary and the Bakry-Émery Ricci curvature tensor.
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 53C23
Mathematics Subject Classification ID: 58J35
Mathematics Subject Classification ID: 35K99
Mathematics Subject Classification ID: 35B53
zbMATH DE Number: 7603381
zbMATH Keywords: gradient estimates
zbMATH Keywords: Liouville theorems
zbMATH Keywords: Riemannian manifolds
zbMATH Keywords: Bakry-Émery curvature
zbMATH Keywords: compact boundary
zbMATH Keywords: mean curvature
zbMATH Keywords: smooth metric measure space
reviewed by: Fernando Galaz-García
MaRDI profile type: Publication
full work available at URL: j.geomphys.2022.104670
OpenAlex ID: W4296285994
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DOI: 10.1016/J.GEOMPHYS.2022.104670
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