Global convergence toward traveling fronts in nonlinear parabolic systems with a gradient structure
type: Item
instance of: scholarly article
title: Global convergence toward traveling fronts in nonlinear parabolic systems with a gradient structure
zbMATH Open document ID: 1152.35047
author: Emmanuel Risler
published in: Annales de l'Institut Henri Poincaré. Analyse Non Linéaire
publication date: 2008-03-31T00:00:00Z
full work available at URL: https://eudml.org/doc/78795
review text: The author considers the nonlinear parabolic system of the form \[ u_t=-\nabla V\left( u\right) +u_{xx} \] where \(u=\left( u_1,\dots,u_n\right) \in \mathbb{R}^n,n\geq 1,x\in \mathbb{R},\) the potential \(V:\mathbb{R}^n\rightarrow \mathbb{R}\) is of class \(C^k\), \(k\geq 3,V\left( u\right) \rightarrow +\infty ,\) when \(\left\| u\right\| \rightarrow +\infty .\) The author proves a global convergence toward bistable fronts which states the invasion of a stable homogeneous equilibrium (a local minimum of the potential) necessarily occurs via traveling front connecting to another equilibrium. The proof is based only on energy methods and it doesn't use any comparison principle.
reviewed by: Cristian Chifu
Mathematics Subject Classification ID: 35K45
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 35B50
Mathematics Subject Classification ID: 35B35
zbMATH DE Number: 5255881
zbMATH Keywords: nonlinear parabolic system
zbMATH Keywords: gradient structure
zbMATH Keywords: global convergence
zbMATH Keywords: bistable traveling front
zbMATH Keywords: energy functional
zbMATH Keywords: maximum principle
MaRDI profile type: Publication
OpenAlex ID: W2024843078
cites work: Q4288396
cites work: Front propagation in periodic excitable media
cites work: Multi-dimensional travelling-wave solutions of a flame propagation model
cites work: Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations
cites work: Bounded, locally compact global attractors for semilinear damped wave equations on \(\mathbb{R}^ N\)
cites work: Long time behavior of solutions of bistable nonlinear diffusion equations
cites work: The approach of solutions of nonlinear diffusion equations to travelling front solutions
cites work: A phase plane discussion of convergence to travelling fronts for nonlinear diffusion
cites work: Q4762886
cites work: Global stability of travelling fronts for a damped wave equation with bistable nonlinearity
cites work: A variational proof of global stability for bistable travelling waves.
cites work: Energy flow in formally gradient partial differential equations on unbounded domains
cites work: The Cauchy problem in local spaces for the complex Ginzburg-Landau equation. II: Contraction methods
cites work: Geometric theory of semilinear parabolic equations
cites work: Convergence of global and bounded solutions of the wave equation with linear dissipation and analytic nonlinearity
cites work: The Cauchy problem for quasi-linear symmetric hyperbolic systems
cites work: The stable, center-stable, center, center-unstable, unstable manifolds
cites work: The complex Ginzburg - Landau equation on large and unbounded domains: sharper bounds and attractors
cites work: Attractors for modulation equations on unbounded domains-existence and comparison
cites work: A global variational structure and propagation of disturbances in reaction-diffusion systems of gradient type
cites work: Monotonicity and convergence results in order-preserving systems in the presence of symmetry
cites work: Stability analysis in order-preserving systems in the presence of symmetry
cites work: Eventual monotonicity and convergence to travelling fronts for the solutions of parabolic equations in cylinders
cites work: Multidimensional traveling wavefronts in a model from combustion theory and in related problems
cites work: Construction of the Leray-Schauder degree for elliptic operators in unbounded domains
cites work: Q4711172
DOI: 10.1016/J.ANIHPC.2006.12.005
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description: scientific article