Existence of regular solutions for semilinear parabolic integro- differential equations
type: Item
instance of: scholarly article
title: Existence of regular solutions for semilinear parabolic integro- differential equations
zbMATH Open document ID: 0606.45005
DOI: 10.1007/BF01760809
author: Hans Engler
published in: Annali di Matematica Pura ed Applicata. Serie Quarta
publication date: 1986-01-01T00:00:00Z
review text: The author studies existence and regularity problems for the semilinear parabolic equation \[ (1)\quad u_ t-\Delta_ xu+a*g(u)=f,\quad (x,t)\in \Omega \times [0,T] \] (2) \(u\equiv 0\), (x,t)\(\in \partial \Omega \times [0,T]\). Here equation (1) can be considered as an extension of the equation \[ (3)\quad u_ t-\Delta_ xu+g(u)=f,\quad (x,t)\in \Omega \times [0,T] \] with boundary condition (2). The crucial growth restriction on g that the author needs to establish the existence of a solution of (1), coincides with the weakest known properties of g that guarantee the existence of a solution of (3). Furthermore, the equation (4) \(u_ t-\Delta_ xu=f\), \((x,t)\in \Omega [0,T]\) with boundary condition (5) \(\partial_ vu+a*g(u)=h\), \((x,t)\in \partial \Omega \times [0,T]\) is studied. The boundary condition (4) is an extension of the condition \(\partial_ vu+g(u)=h\), \((x,t)\in \partial \Omega \times [0,T].\) For the problem (4) and (5) the regularizing effect (in time) of the integral operator is also used. Both scalar and vector equations are considered.
Mathematics Subject Classification ID: 45K05
zbMATH DE Number: 3979749
zbMATH Keywords: Dirichlet boundary conditions
zbMATH Keywords: heat equation
zbMATH Keywords: convolution
zbMATH Keywords: growth restriction
zbMATH Keywords: global solutions
zbMATH Keywords: existence
zbMATH Keywords: regularity
zbMATH Keywords: semilinear parabolic equation
reviewed by: Gustaf Gripenberg
MaRDI profile type: Publication
cites work: Nonlinear second order equations with applications to partial differential equations
cites work: A Version of the Chain Rule and Integrodifferential Equations in Hilbert Spaces
cites work: Q3326145
cites work: Stabilization of solutions for a class of parabolic integro-differential equations
cites work: Regularity properties for the heat equation subject to nonlinear boundary constraints
cites work: Q3924584
cites work: Q5630031
cites work: On some integral and integro-differential equations in a Hilbert space
cites work: Sommes d'opérateurs linéaires et équations différentielles opérationnelles
cites work: An Abstract Parabolic Volterra Integrodifferential Equation
cites work: Global solutions of semilinear evolution equations satisfying an energy inequality
cites work: Q3878026
cites work: Q5562267
cites work: Q5724127
cites work: Q3322626
cites work: Q3894443
cites work: Globale klassische Lösungen des Cauchyproblems semilinearer parabolischer Differentialgleichungen
cites work: Uniform bounds from bounded \(L_ p-\)functionals in reaction-diffusion equations
cites work: Existence and Uniqueness of Solutions of Second Order Nonlinear Differential Equations
cites work: Q5601742
cites work: Some theorems on the estimate and existence of solutions of integro-differential equations of parabolic type
cites work: Starke Lösungen semilinearer parabolischer Differentialgleichungen
cites work: Neue Resolventenabschätzungen für elliptische Differentialoperatoren und semilineare parabolische Gleichungen
cites work: An Abstract Semilinear Volterra Integrodifferential Equation
cites work: Quantitative maximum principles and strongly coupled gradient-like reaction-diffusion systems
cites work: Regularity and asymptotic behavior for the second order parabolic equation with nonlinear boundary conditions
description: scientific article; zbMATH DE number 3979749