Asymptotic states for equations of reaction and diffusion

Fife, Paul C.

doi:10.1090/s0002-9904-1978-14502-9

Cited by 100 publications

(45 citation statements)

References 62 publications

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“…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%

“…Thus we shall be concerned with solutions of equation (1.1) of the form u = f (x − σt) where f (ξ) is defined and monotonic for −∞ < ξ < ∞, and where f (ξ) → ± as ξ → ±∞ for some ± ∈ I for which c( ± ) = 0 and + = − . Such solutions connecting two equilibrium states of the equation have long been of interest [25,53,90,93,172,192,267,268]. Without any loss of generality we shall take − = < ∞ and + = 0, and, in line with Definition 4, term the solution a wavefront from to 0.…”

Section: Wavefrontsmentioning

confidence: 99%

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Travelling Waves in Nonlinear Diffusion-Convection Reaction

Gilding¹,

Kersner²

2004

170

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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.

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

confidence: 99%

Section: Wavefrontsmentioning

confidence: 99%

Travelling Waves in Nonlinear Diffusion-Convection Reaction

Gilding¹,

Kersner²

2004

170

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“…The asymptotic stability of stationary periodic solutions for the Fisher equation was proved in paper [7]. Periodic problem for the Hopf equation was studied in paper [15].…”

Section: Introductionmentioning

confidence: 99%

Asymptotics of solutions to the periodic problem for the nonlinear damped wave equation

Hayashi

Naumkin

Rodriguez-Ceballos

2010

Nonlinear Differ. Equ. Appl.

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Abstract.We study large time asymptotic behavior of solutions to the periodic problem for the nonlinear damped wave equationwhere Ω = [−π, π] , α, β, λ, σ > 0. We prove that if the initial data φ ∈ H 1 and ψ ∈ L 2 , then there exists a unique solution u (t, x) ∈ C [0, ∞) ; H 1 of the periodic problem which has the time decay estimatesfor all t > 0. Moreover under some additional conditions we find the asymptotic formulas for the solutions. Mathematics Subject Classification (2000). 35L15, 35K20.

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“…

446BOOK REVIEWS the predecessor volume [6] at hand. Many further results are given as exercises, with solutions or references.

Besides, I found no serious lacuna in the index and very few misprints in the book, none of them bothersome.

…”

mentioning

confidence: 99%

“…BOOK REVIEWS the predecessor volume [6] at hand. Many further results are given as exercises, with solutions or references.…”

mentioning

confidence: 99%

Book Review: Traveling wave solutions of parabolic systems

Gardner¹

1995

Bull. Amer. Math. Soc.

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446BOOK REVIEWS the predecessor volume [6] at hand. Many further results are given as exercises, with solutions or references.Besides, I found no serious lacuna in the index and very few misprints in the book, none of them bothersome. (At the author's request, I mention that the remark on page 274 should be deleted.) As in [5 and 6], for which Helgason received the 1988 Steele Prize for expository writing, the style is very fluent and pleasant, conducting the reader at a regular pace. I think the present book will be a most valuable (and reasonably priced) reference for anyone interested in Radon transforms and analysis on semisimple Lie groups. 4. V.

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Asymptotic states for equations of reaction and diffusion

Cited by 100 publications

References 62 publications

Travelling Waves in Nonlinear Diffusion-Convection Reaction

Travelling Waves in Nonlinear Diffusion-Convection Reaction

Asymptotics of solutions to the periodic problem for the nonlinear damped wave equation

Book Review: Traveling wave solutions of parabolic systems

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