Behavior of solutions to nonlinear parabolic equations at large time
type: Item
instance of: scholarly article
title: Behavior of solutions to nonlinear parabolic equations at large time
zbMATH Open document ID: 0863.35051
DOI: 10.1007/BF02109832
author: T. I. Zelenyak
author: Mikhail P. Vishnevskii
author: Mikhail M. jun. Lavrentiev
published in: Siberian Mathematical Journal
publication date: 1996-09-03T00:00:00Z
review text: Initial-boundary value problems for systems of quasilinear parabolic equations are considered. The couple of the equations is assumed to be cooperative, so that the resulting semiflow falls in the class of strongly monotone dynamical systems. The latter means that any two pointwise and componentwise ordered initial conditions give solutions that are ordered at any time instant. The author presents several theorems on existence of stable and unstable equilibria, domains of attraction, existence of monotone heteroclinic solutions, eventual monotonicity and stabilization of solutions. The results presented overlap several theorems in the existing literature. Some of them are quoted in the paper under review, for others see the monographs [\textit{P. Hess}, Periodic-parabolic boundary value problems and positivity, Pitman Res. Notes 247, Longman, Harlow (1991; Zbl 0731.35050)] and [\textit{H. L. Smith}, Monotone dynamical systems, American Mathematical Society, Providence, RI (1995; Zbl 0821.34003)].
Mathematics Subject Classification ID: 35K60
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35B05
Mathematics Subject Classification ID: 35B35
zbMATH DE Number: 923517
zbMATH Keywords: Lyapunov functional
zbMATH Keywords: quasilinear parabolic equations
zbMATH Keywords: strongly monotone dynamical systems
zbMATH Keywords: domains of attraction
zbMATH Keywords: monotone heteroclinic solutions
reviewed by: Peter Poláčik
MaRDI profile type: Publication
cites work: Q5630074
cites work: Q3963437
cites work: Convergence of solutions of one-dimensional semilinear parabolic equations
cites work: Asymptotic behavior and stability of solutions of semilinear diffusion equations
cites work: Q3961188
cites work: The zero set of a solution of a parabolic equation.
cites work: Q3872868
cites work: Q5640108
cites work: Singular minimizers for regular one-dimensional problems in the calculus of variations
cites work: Stability and convergence in strongly monotone dynamical systems.
cites work: Convergence in smooth strongly monotone flows defined by semilinear parabolic equations
cites work: User’s guide to viscosity solutions of second order partial differential equations
cites work: Q4728730
cites work: Connecting orbits in scalar reaction diffusion equations. II: The complete solution
cites work: A Nonconforming Finite-Element Method for the Two-Dimensional Cahn–Hilliard Equation
cites work: Slow motion for the Cahn-Hilliard equation in one space dimension
cites work: Inertial manifolds and inertial sets for the phase-field equations
cites work: Q4893302
cites work: Convergence in general periodic parabolic equations in one space dimension
cites work: A Poincaré-Bendixson theorem for scalar reaction diffusion equations
cites work: Heteroclinic orbits of semilinear parabolic equations
full work available at URL: https://doi.org/10.1007/bf02109832
OpenAlex ID: W1969843688
Recommended article: Large time behavior of a solution of a parabolic equation
Recommended article: Large time behavior of solutions to the nonlinear pseudo-parabolic equation
Recommended article: Large time behavior of solutions for parabolic equations with nonlinear gradient terms
Recommended article: Large-time behavior of smooth solutions to a nonuniformly parabolic equation
Recommended article: On the behaviour of solutions of parabolic equations for large values of time
Recommended article: Large time behavior of ODE type solutions to a nonlinear parabolic system
Recommended article: Q4286455
Recommended article: Long-time behaviour of solutions of a class of nonlinear parabolic equations
Recommended article: Large time behavior for some nonlinear degenerate parabolic equations
Recommended article: Q3708397
description: scientific article; zbMATH DE number 923517