Validity of the Linear Speed Selection Mechanism for Fronts of the Nonlinear Diffusion Equation

Benguria, Rafael D.; Depassier, M. C.

doi:10.1103/physrevlett.73.2272

Cited by 46 publications

(36 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For the nonlinear selection cases, c min > c lin , Rothe [Rot81] and Roquejoffre [Roq97] proved that, if the initial data decays sufficiently rapidly, the large-time solutions to (1.4) not only propagate with speed c min but also approach the profile of a travelling wave with minimal speed. Whereas the connection between large-time behaviour and existence of travelling waves for monostable equations is satisfactorily resolved, the distinction between nonlinear or linear selection is still a challenging question, see for example [vS89], [EvS00], [BD94]. Only a few rigorous results for general monostable nonlinearities are available.…”

mentioning

confidence: 99%

Travelling-wave analysis of a model describing tissue degradation by bacteria

Hilhorst¹,

King²,

Röger³

2007

Eur. J. Appl. Math

View full text Add to dashboard Cite

Abstract. We study travelling-wave solutions for a reaction-diffusion system arising as a model for host-tissue degradation by bacteria. This system consists of a parabolic equation coupled with an ordinary differential equation. For large values of the 'degradation-rate parameter' solutions are well approximated by solutions of a Stefan-like free boundary problem, for which travelling-wave solutions can be found explicitly. Our aim is to prove the existence of travelling waves for all sufficiently large wave-speeds for the original reaction-diffusion system and to determine the minimal speed. We prove that for all sufficiently large degradation rates the minimal speed is identical to the minimal speed of the limit problem. In particular, in this parameter range, nonlinear selection of the minimal speed occurs.

show abstract

mentioning

confidence: 99%

Travelling-wave analysis of a model describing tissue degradation by bacteria

Hilhorst¹,

King²,

Röger³

2007

Eur. J. Appl. Math

View full text Add to dashboard Cite

show abstract

“…Theorem 37 expands on the work of Benguria and Depassier [33,[35][36][37][38] and of Benguria, Cisternas and Depassier [32]. Indeed, the most essential ideas behind the proof of part (a) can be found in [32,33,35,36,38] and part (b) in [37].…”

Section: Elementary Analysis Shows That This Occurs If and Only Ifmentioning

confidence: 95%

“…Indeed, the most essential ideas behind the proof of part (a) can be found in [32,33,35,36,38] and part (b) in [37].…”

Section: Elementary Analysis Shows That This Occurs If and Only Ifmentioning

confidence: 99%

“…In other words, it is possible to view the integral equation (1.9) as a description of travelling-wave solutions of (1.1) in their phase-space. The relationship between travelling-wave solutions of equations of the type (1.1) and the ordinary differential equation (1.11) has been noted and utilized by previous authors [20][21][22][23][32][33][34][35][36][37][38]61,90,93,96,97,159,162,172,191,192,213,222,246,247,250,254,266,268,270,282], and for a related problem in [153,154,282].…”

Section: C(r)a (R) G(r) Drmentioning

confidence: 99%

See 1 more Smart Citation

Travelling Waves in Nonlinear Diffusion-Convection Reaction

Gilding¹,

Kersner²

2004

170

View full text Add to dashboard Cite

The study of travelling waves or fronts has become an essential part of the mathematical analysis of nonlinear diffusion-convection-reaction processes. Whether or not a nonlinear second-order scalar reaction-convection-diffusion equation admits a travelling-wave solution can be determined by the study of a singular nonlinear integral equation. This article is devoted to demonstrating how this correspondence unifies and generalizes previous results on the occurrence of travelling-wave solutions of such partial differential equations. The detailed comparison with earlier results simultaneously provides a survey of the topic. It covers travelling-wave solutions of generalizations of the Fisher, Newell-Whitehead, Zeldovich, KPP and Nagumo equations, the Burgers and nonlinear Fokker-Planck equations, and extensions of the porous media equation.

show abstract

“…for which rigorous results can be obtained [3,4,6,5,[7][8][9][10][11]. Depending on the situation being considered the reaction term f (u) satisfies additional properties.…”

Section: Introductionmentioning

confidence: 99%

Counterexample to a conjecture of Goriely for the speed of fronts of the reaction-diffusion equation

Cisternas

Depassier

1997

Phys. Rev. E

Self Cite

View full text Add to dashboard Cite

In a recent paper Goriely considers the one-dimensional scalar reactiondiffusion equation u t = u xx + f (u) with a polynomial reaction term f (u) and conjectures the existence of a relation between a global resonance of the hamiltonian system u xx +f (u) = 0 and the asymptotic speed of propagation of fronts of the reaction diffusion equation. Based on this conjecture an explicit expression for the speed of the front is given. We give a counterexample to this conjecture and conclude that additional restrictions should be placed on the reaction terms for which it may hold. 82.40.Ck, 47.10+g, 3.40Kf, 2.30.Hq

show abstract

Validity of the Linear Speed Selection Mechanism for Fronts of the Nonlinear Diffusion Equation

Cited by 46 publications

References 10 publications

Travelling-wave analysis of a model describing tissue degradation by bacteria

Travelling-wave analysis of a model describing tissue degradation by bacteria

Travelling Waves in Nonlinear Diffusion-Convection Reaction

Counterexample to a conjecture of Goriely for the speed of fronts of the reaction-diffusion equation

Contact Info

Product

Resources

About