Strong full bounded solutions of nonlinear parabolic equations with nonlinear boundary conditions
type: Item
instance of: scholarly article
title: Strong full bounded solutions of nonlinear parabolic equations with nonlinear boundary conditions
zbMATH Open document ID: 1223.35205
author: M. N. Nkashama
author: Nsoki Mavinga
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2011-07-19T00:00:00Z
review text: The authors deal with the following nonlinear parabolic boundary value problem \[ \begin{cases} \frac{\partial u}{\partial t}(x,t)-Lu(x,t)= f(x,t,u) &\text{a.e. in }\Omega \times\mathbb R,\\ {\mathcal B}u=\varphi(x,t,u) &\text{a.e. on }\partial \Omega \times \mathbb R,\\ \sup_{\Omega \times\mathbb R}| u(x,t)| <\infty, \end{cases} \] where \(\Omega \) is a bounded, open and connected subset of \(\mathbb R^N\), \(L\) is a second-order, uniformly elliptic differential operator with time-dependent coefficients, and \({\mathcal B}\) is a linear first-order boundary operator which is of either Dirichlet, Neumann, or regular oblique type. It is supposed that the coefficients of the operators \(L\) and \({\mathcal B}\) are measurable and bounded. The reaction and the boundary nonlinearities \(f\) and \(\varphi \) are Carathéodory functions. The authors are interested in bounded solutions existing for all times. The authors show first, by a counter-example, that the weak maximum principle fails when solutions exits for all times, even when the coefficients in the equation are very smooth. They consider a boundedness condition at minus infinity in order to establish one-sided \(L^\infty\) a priori estimates for the solutions to linear boundary value problems and derive a weak maximum principle which is valid on the entire real line in time. This counter-example shows that such a condition is essential for the maximum or comparison principle to hold. The authors formulate the general assumptions and state the main result concerning the existence, and, in some instance, positivity and uniqueness of bounded solutions existing for all times for nonlinear problems with nonlinear boundary conditions. The existence of minimal or maximal solutions is also discussed requiring only that the reaction nonlinearity \(f\) is Carathéodory, no one-sided Lipschitz nor Hölder continuity are required on \(f\). They prove the main results using some auxiliary results, comparison techniques, Gali…
reviewed by: Cristian Chifu
Mathematics Subject Classification ID: 35K61
Mathematics Subject Classification ID: 35K58
Mathematics Subject Classification ID: 35B45
Mathematics Subject Classification ID: 35B50
zbMATH DE Number: 5925810
zbMATH Keywords: subsolutons
zbMATH Keywords: supersolutions
zbMATH Keywords: interpolation inequalities
zbMATH Keywords: nonlinear approximation
zbMATH Keywords: generalized full bounded solutions
zbMATH Keywords: minimal solutions
zbMATH Keywords: maximal solutions
MaRDI profile type: Publication
full work available at URL: j.na.2011.05.012
OpenAlex ID: W2014716992
cites work: Q4550348
cites work: Solutions of parabolic boundary problems existing for all time
cites work: Q3682852
cites work: Q4347837
cites work: Q5613838
cites work: A maximum principle for bounded solutions of the telegraph equation in space dimension three
cites work: Q3998030
cites work: Unstable periodic solutions of a parabolic problem in the presence of non-well-ordered lower and upper solutions
cites work: Q4716216
cites work: Convergence in almost periodic Fisher and Kolmogorov models
cites work: Q4444862
cites work: Q4933752
cites work: Q4272980
cites work: Q5624091
cites work: Upper and lower solutions and semilinear second order elliptic equations with non-linear boundary conditions
cites work: Corrigendum: Upper and lower solutions and semilinear second order elliptic equations with non-linear boundary conditions
cites work: Q5533735
cites work: Q4002481
cites work: Q4354424
cites work: Q5520962
cites work: Elliptic partial differential equations
cites work: Q3315877
cites work: Q5562267
cites work: Q4748596
cites work: Q4929960
cites work: Q4116725
cites work: Equivalent Norms for Sobolev Spaces
cites work: Parabolic differential inequalities and Chaplighin's method
cites work: Q3901719
DOI: 10.1016/J.NA.2011.05.012
description: scientific article; zbMATH DE number 5925810