Extinction of superdiffusions and semilinear partial differential equations
type: Item
instance of: scholarly article
title: Extinction of superdiffusions and semilinear partial differential equations
zbMATH Open document ID: 0938.35059
author: K. Appert
published in: Journal of Functional Analysis
publication date: 2000-06-12T00:00:00Z
review text: Let \(L\) be a uniformly elliptic second order differential operator in \(\mathbb{R}^d\) with bounded locally Hölder continuous coefficients and let \(\psi\) be a function from \(\mathbb{R}^d\times \mathbb{R}_+\) to \(\mathbb{R}_+\). A measure-valued branching process in \(\mathbb{R}^d\) associated with the nonlinear operator \(Lu-\psi(x,u)\) is called \((L,\psi)\)-superdiffusion. One says that a regular domain \(D\) in \(\mathbb{R}^d\) is special if there exists a strictly positive solution \(u\) of the equation \(Lu=\psi(x,u)\) which is equal to \(0\) on \(\partial D.\) Under the assumption that the function \(L1\) is bounded, the authors describe the class of special domains \(D\) in terms of the extinction time of \((L,\psi)\)-superdiffusion in \(D\). For elliptic operators \(L\) which can be represented in the divergence form, the authors show that a bounded smooth domain \(D\) is special if and only if the principal eigenvalue \(\lambda_1(D)\) of \(-L\) in \(D\) is negative. (In this case, the description of the class of special domains depends only on \(L\) but not on \(\psi\).) The case of the Beltrami-Laplace operator on a complete Riemannian manifold is covered.
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 35B32
Mathematics Subject Classification ID: 35B05
zbMATH DE Number: 1274804
zbMATH Keywords: Hölder continuous coefficients
zbMATH Keywords: measure-valued branching process
zbMATH Keywords: extinction time
zbMATH Keywords: Beltrami-Laplace operator
zbMATH Keywords: second order
zbMATH Keywords: semilinear
zbMATH Keywords: \(p\)-Laplacian
zbMATH Keywords: variational solution
MaRDI profile type: Publication
OpenAlex ID: W2045289349
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DOI: 10.1006/JFAN.1998.3355
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description: scientific article; zbMATH DE number 1274804