Analytic semigroups, degenerate elliptic operators and applications to nonlinear Cauchy problems
type: Item
instance of: scholarly article
title: Analytic semigroups, degenerate elliptic operators and applications to nonlinear Cauchy problems
zbMATH Open document ID: 0709.35065
DOI: 10.1007/BF01765950
author: Vincenzo Vespri
published in: Annali di Matematica Pura ed Applicata. Serie Quarta
publication date: 1989-01-01T00:00:00Z
review text: The nonlinear problem \[ u_ t=\phi (\Theta (u))\Delta (\chi (u)),\quad x\text{ in } {\bar \Omega},\quad t\geq 0;\quad u(x,0)=u_ 0(x) \] is considered, where \(\Omega\) is a bounded domain in \({\mathbb{R}}^ n\) with \({\mathbb{C}}^{\infty}\) boundary. \(\phi\), \(\Theta\), \(\chi\) are smooth functions and \(u_ 0\) is positive and sufficiently regular. A large class of problems in applications is of this form. He studies the problem by analyzing the associated degenerate elliptic problem \((\lambda -E)u=f\), \(\lambda\in {\mathbb{C}}.\) He proves \(L^ p(\Omega)\) and \(C^{\alpha}({\bar \Omega})\)-norm estimates for the solution of the elliptic problem.
Mathematics Subject Classification ID: 35K60
Mathematics Subject Classification ID: 35J70
Mathematics Subject Classification ID: 35B45
Mathematics Subject Classification ID: 47D06
zbMATH DE Number: 4165485
zbMATH Keywords: nonlinear heat equation
zbMATH Keywords: \(L^ p\)-estimates
zbMATH Keywords: \(C^{\alpha }\)-estimates
zbMATH Keywords: associated degenerate elliptic problem
reviewed by: Robert L. Taylor
MaRDI profile type: Publication
cites work: On the pointwise behavior of the solutions of the porous medium equation as t approaches zero or infinity
cites work: Persistence and extinction in single-species reaction-diffusion models
cites work: Local existence and regularity for a class of degenerate parabolic equations
cites work: How an Initially Stationary Interface Begins to Move in Porous Medium Flow
cites work: Conformal deformations of complete manifolds with negative curvature
cites work: Régularité et théorie spectrale pour une classe d'opérateurs elliptiques dégénérées
cites work: Régularité analytique et iterés d'opérateurs elliptiques dégénérées; applications
cites work: Q4075342
cites work: Complete Metrics Conformal to the Hyperbolic Disc
cites work: Q3882115
cites work: Q3900391
cites work: Analytic semigroups generated by non-variational elliptic systems of second order under Dirichlet boundary conditions
cites work: Analytic semigroups generated on Hölder spaces by second order elliptic systems under Dirichlet boundary conditions
cites work: An Upper Bound for the Waiting Time for Nonlinear Degenerate Parabolic Equations
cites work: Q4727718
cites work: Q4001895
cites work: A degenerate diffusion problem not in divergence form
cites work: Equations d'évolution abstraites non linéaires de type parabolique
cites work: On second order differential operators
cites work: Generalized second order differential operators and their lateral conditions
cites work: Blow-up of solutions of nonlinear degenerate parabolic equations
cites work: Singular nonlinear parabolic boundary value problems in one space dimension
cites work: Q3796077
cites work: Highly degenerate parabolic boundary value problems
cites work: Q5617095
cites work: The Porous Medium Equation in One Dimension
cites work: Q3276084
cites work: Interpolation Spaces Between Domains of Elliptic Operators and Spaces of Continuous Functions with Applications to Nonlinear Parabolic Equations
cites work: Q3790167
cites work: Generation of Analytic Semigroups by Strongly Elliptic Operators
cites work: Q3262584
Recommended article: Analytic semigroups generated by elliptic operators in \(L^ 1\) and parabolic equations
Recommended article: Analytic semigroups on \(H^1(\Omega)\) generated by degenerate elliptic operators
Recommended article: Q4728870
Recommended article: Generation of analytic semigroups by degenerate elliptic operators
Recommended article: Analytic semigroups generated in \(L^p\) by elliptic operators with high order degeneracy at the boundary
Recommended article: Analytic semigroups and degenerate elliptic operators with unbounded coefficients: A probabilistic approach
Recommended article: Q5462247
Recommended article: Q4863373
Recommended article: Analytic semigroups and semilinear initial boundary value problems
Recommended article: Degenerate Nonlinear Semigroups of Operators and Their Applications
description: scientific article; zbMATH DE number 4165485