Existence of traveling waves of invasion for Ginzburg-Landau-type problems in infinite cylinders
type: Item
instance of: scholarly article
title: Existence of traveling waves of invasion for Ginzburg-Landau-type problems in infinite cylinders
zbMATH Open document ID: 1165.35028
published in: Archive for Rational Mechanics and Analysis
publication date: 2008-06-30T00:00:00Z
review text: This paper is concerned with the existence of traveling wave solutions of reaction-diffusion systems of the gradient type \(u_t=\Delta u-\nabla_uV(u)\), where \(u=u(x,t)\). After introducing the functional framework, the authors define the main variational problem associated to this nonlinear parabolic system. Next, it is established the existence of solutions, viewed as minimizers of the associated energy functional. A particular attention is given to the boundedness and regularity of solutions, as well as to global gradient estimates for the minimizers. Uniform estimates allow the authors to obtain information on the exponential decay of functions obeying related estimates of the norm of solutions. In the last part of the present paper there are deduced several criteria of existence and non-existence of traveling waves. There are also established several properties of minimizers, such as their one-dimensionality in the case of Neumann boundary conditions, or the fact that for potentials \(V\) depending only on the magnitude of the vector \(u\), the minimizers are scalar (up to a constant vector). Several applications of the abstract results by using a computer-assisted approach conclude the present paper.
reviewed by: Vicenţiu D. Rădulescu
Mathematics Subject Classification ID: 35K57
Mathematics Subject Classification ID: 35A05
Mathematics Subject Classification ID: 35K60
Mathematics Subject Classification ID: 35A15
Mathematics Subject Classification ID: 35B40
zbMATH DE Number: 5294564
zbMATH Keywords: Ginzburg-Landau system
zbMATH Keywords: traveling wave
zbMATH Keywords: reaction-diffusion equation
author: Marcello Lucia
author: C. B. Muratov
author: M. Novaga
MaRDI profile type: Publication
full work available at URL: s00205-007-0097-x
OpenAlex ID: W2164503142
cites work: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I
cites work: Q4089232
cites work: Variational characterization of the speed of propagation of fronts for the nonlinear diffusion equation
cites work: Pattern propagation in nonlinear dissipative systems
cites work: Travelling fronts in cylinders
cites work: Phase plane analysis of one-dimensional reaction diffusion waves with degenerate reaction terms
cites work: Q4224212
cites work: Q2736008
cites work: Pattern formation outside of equilibrium
cites work: An introduction to \(\Gamma\)-convergence
cites work: Shape analysis via oriented distance functions
cites work: Glauber evolution with Kac potentials. I. Mesoscopic and macroscopic limits, interface dynamics
cites work: Q3134551
cites work: Large patterns of elliptic systems in infinite cylinders
cites work: Q4189096
cites work: The approach of solutions of nonlinear diffusion equations to travelling front solutions
cites work: Q5187597
cites work: Travelling fronts in nonlinear diffusion equations
cites work: Q4039920
cites work: Q4692812
cites work: Wave solutions to reaction-diffusion systems in perforated domains
cites work: Variational principles for propagation speeds in inhomogeneous media
cites work: Q3802757
cites work: Linear vs. nonlinear selection for the propagation speed of the solutions of scalar reaction-diffusion equations invading an unstable equilibrium
cites work: Analytic semigroups and optimal regularity in parabolic problems
cites work: Travelling wave solutions of the degenerate KolmogorovPetrovskiPiskunov equation
cites work: Essential manifolds for an elliptic problem in an infinite strip
cites work: Homoclinic orbits in Hamiltonian systems and heteroclinic orbits in gradient and gradient-like systems
cites work: A global variational structure and propagation of disturbances in reaction-diffusion systems of gradient type
cites work: Q3994920
cites work: Travelling Wave Solutions to a Gradient System
cites work: Eventual monotonicity and convergence to travelling fronts for the solutions of parabolic equations in cylinders
cites work: Global Stability of Traveling Fronts and Convergence Towards Stacked Families of Waves in Monotone Parabolic Systems
cites work: Q4746129
cites work: Partial differential equations. 3: Nonlinear equations
cites work: Infinitely Many Traveling Wave Solutions of a Gradient System
cites work: Front propagation into unstable states
cites work: The asymptotic behavior of the solutions of some semilinear elliptic equations in cylindrical domains
cites work: EXISTENCE OF MULTIDIMENSIONAL TRAVELLING WAVES AND SYSTEMS OF WAVES
cites work: Propagation of frontal polymerization—crystallization waves
DOI: 10.1007/S00205-007-0097-X
description: scientific article; zbMATH DE number 5294564