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2004 | Buch

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Travelling Waves in Nonlinear Diffusion-Convection Reaction

herausgegeben von: Brian H. Gilding, Robert Kersner

Verlag: Birkhäuser Basel

Buchreihe : Progress in Nonlinear Differential Equations and Their Applications

Enthalten in: Professional Book Archive

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This monograph has grown out of research we started in 1987, although the foun dations were laid in the 1970's when both of us were working on our doctoral theses, trying to generalize the now classic paper of Oleinik, Kalashnikov and Chzhou on nonlinear degenerate diffusion. Brian worked under the guidance of Bert Peletier at the University of Sussex in Brighton, England, and, later at Delft University of Technology in the Netherlands on extending the earlier mathematics to include nonlinear convection; while Robert worked at Lomonosov State Univer sity in Moscow under the supervision of Anatolii Kalashnikov on generalizing the earlier mathematics to include nonlinear absorption. We first met at a conference held in Rome in 1985. In 1987 we met again in Madrid at the invitation of Ildefonso Diaz, where we were both staying at 'La Residencia'. As providence would have it, the University 'Complutense' closed down during this visit in response to student demonstra tions, and, we were very much left to our own devices. It was natural that we should gravitate to a research topic of common interest. This turned out to be the characterization of the phenomenon of finite speed of propagation for nonlin ear reaction-convection-diffusion equations. Brian had just completed some work on this topic for nonlinear diffusion-convection, while Robert had earlier done the same for nonlinear diffusion-absorption. There was no question but that we bundle our efforts on the general situation.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction

Abstract

Wave phenomena are observed in many natural reaction, convection and diffusion processes. This alone is motivation for studying their occurrence. Other reasons why the study of travelling-wave solutions has become such an essential part of the mathematical analysis of nonlinear diffusion-convection-reaction processes are that: the analysis of travelling waves provides a means of finding explicit solutions of the equation; in general travelling-wave solutions are easier to analyse and therewith discern properties to be expected of other solutions; such solutions can be used as tools in comparison principles and the like to determine the properties of general solutions; and, last but not least, in conformance with their natural occurrence in many mathematically modelled phenomena, they characterize the long-term behaviour in numerous situations.

Brian H. Gilding, Robert Kersner

Chapter 2. General theory

Abstract

Motivated by the modelling origins of equation (1.1), where the unknown is generally nonnegative, we shall henceforth assume that the coefficients of the equation are defined in some closed interval I with minimum 0. However, because we would like to deal with both bounded and unbounded solutions in a single framework, let us provisionally denote the supremum of this interval by ℓ which may be finite or infinite Thus,

$$ I = [0,\ell ) with \ell = \infty , or, I = [0,\ell ) with 0 < \ell < \infty . $$

Brian H. Gilding, Robert Kersner

Chapter 3. Transformations

Abstract

There are a number of transformations known, through which the existence of a wavefront solution of one reaction-convection-diffusion equation can be used to determine the existence of a wavefront solution of another. These transformations become quite transparent in terms of the integral equation (1.9). In fact, they are embodied in the following two theorems.

Brian H. Gilding, Robert Kersner

Chapter 4. Travelling waves

Abstract

In this chapter we report a number of general results on the existence of semiwavefront solutions of equation (1.1) which can be obtained from the study of the integral equation (1.9).

Brian H. Gilding, Robert Kersner

Chapter 5. Convection-diffusion

Abstract

Considering only diffusion-convection processes, i.e. the equation

$$ u_t = (a(u))_{xx} + (b(u))_{x}, $$

(5.1)

the integral equation (1.9) reduces to the simple identity θ (s) = σ s + b(s). Moreover, by Lemma 2.40 any such ‘solution’θ satisfies the integrability condition in an interval [0, δ) if and only if it is positive in (0, δ ). The search for nonnegative solutions of the integral equation satisfying the integrability condition is therefore reduced to the search for a- such that σ s + b(s) > 0 for all 0 < s < δ for some 0 < δ ≤ ℓ. This leads readily to the next result.

Brian H. Gilding, Robert Kersner

Chapter 6. Reaction-diffusion

Abstract

The class of equations of the type (1.1) in which the convection term is absent, i.e. for which the equation has the form

$$ ut = {\left( {a\left( u \right)} \right)_{xx}} + c\left( u \right) $$

(6.1)

encompasses the Fisher equation, the Newell—Whitehead equation, the Zeldovich equation, the KPP equation, the Nagumo equation, and many other commonly-used models of diffusion-reaction processes. In these models the reaction term does not change sign in a right neighbourhood of zero, and, generally the coefficients a and c are smooth. For an equation of the class (6.1) the corresponding integral equation (1.9) reduces to

$$ \theta \left( s \right) = \sigma s - \int_0^s {\frac{{c(r)a'(r)}}{{\theta (r)}}dr.} $$

(6.2)

Moreover, when c has a fixed sign near zero or when ca’ is sufficiently smooth, this equation possesses a structure which is relatively convenient for analysis. In this chapter, we shall utilize this structure to examine the existence of semi-wavefront solutions of the reaction-diffusion equation (6.1) when c has a definite sign near zero, and, when ca’ is continuously differentiable in a right neighbourhood of zero. Furthermore, we shall identify circumstances under which such a solution is positive everywhere or may have bounded support.

Brian H. Gilding, Robert Kersner

Chapter 7. Power-law equations

Abstract

We turn now to the application of the integral equation (1.9) for the definitive analysis of semi-wavefront solutions for two specific classes of equation (1.1). The first of these is

$$ {u_t} = {\left( {{u^m}} \right)_{xx}} + {b_0}{\left( {{u^n}} \right)_x} + \left\{ {\begin{array}{*{20}{c}} {{c_0}{u^p}foru > 0} \\ {0foru = 0} \end{array}} \right. $$

(7.1)

where m, n p bo and c_o are real parameters. Equations of this type have long been of interest as a tractable prototype for more general equations of the class (1.1). Without the convection and reaction terms, equation (7.1) is simply the porous media equation. The integral equation approach leads to the following characterization of travelling-wave solutions for this category of equations.

Brian H. Gilding, Robert Kersner

Chapter 8. Wavefronts

Abstract

The subject of this and the ensuing two chapters is the existence of wavefront solutions of equations of the class (1.1). 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) → ℓ ^±as ξ → ±∞ for some ℓ ^±∈ I for which c( ℓ ^±)= 0 and ℓ ⁺ ≠ ℓ ^-.Such solutions connecting two equilibrium states of the equation have long been of interest. Without any loss of generality we shall take ℓ ^- = ℓ < ∞ and ℓ ⁺ = 0, and, in line with Definition 2.4, term the solution a wavefront from . ℓ to 0.

Brian H. Gilding, Robert Kersner

Chapter 9. Wavefronts for convection-diffusion

Abstract

As mentioned earlier, when the reaction term in (1.1) is absent and the partial differential equation is

$$ {u_t} = {\left( {a\left( u \right)} \right)_{xx}} + {\left( {b\left( u \right)} \right)_x} $$

(9.1)

the integral equation (1.9) reduces to the simple identity θ(s) = σs+b(s) . Subsequently, if this ‘equation’ is to have a nonnegative solution on [0,ℓ] with ℓ < ∞ such that θ(ℓ)= 0, then necessarily σ = -b (ℓ) / ℓ and θ (s)= b (s) — sb (ℓ) / ℓ ≥ 0 for all 0 ≤ s ≤ℓ . By Lemma 2.40 though θ satisfies the integrability condition if and only if it is positive on (0, ℓ). We conclude the following.

Brian H. Gilding, Robert Kersner

Chapter 10. Wavefronts for reaction-diffusion

Abstract

Throughout this chapter we consider only wavefront solutions of reaction-diffusion equations of the form

$$ ut = {\left( {a\left( u \right)} \right)_{xx}} + c\left( u \right) $$

(10.1)

where a and c satisfy Hypothesis 2.1 with ℓ <∞. Since the pioneering work of Fisher [105] and of Kolmogorov, Petrovskii and Piskunov [186], much attention has been paid to the study of such solutions for equations of this class. The goal of this chapter is to show how the correspondence between travelling-wave solutions of equation (1.1) and solutions of the integral equation (1.9) may be invoked to generalize these earlier results. The theorems of Chapter 6 cover the previous results on semi-wavefront solutions of equations of the type (10.1).

Brian H. Gilding, Robert Kersner

Chapter 11. Unbounded waves

Abstract

One of the motivations for studying travelling-wave solutions of equations of the class (1.1) is that these solutions may be used as a tool for determining the properties of an arbitrary solution of the partial differential equation. As one illustration we mention the characterization of finite speed of propagation using semiwavefront solutions in [126,127]; as another, the characterization of complete and incomplete blow-up in diffusion-reaction processes dependent upon whether or not the equation admits an unbounded strict semi-wavefront solution in [114].

Brian H. Gilding, Robert Kersner

Chapter 12. Wavefronts and unbounded waves for power-law equations

Abstract

With the porous media equation as prototype, equations of the class (1.1) with power-law coefficients have attracted much interest to date. In this chapter, we shall classify all the global monotonic travelling-wave solutions decreasing to 0 and all the unbounded monotonic semi-wavefront solutions decreasing to 0 for equations of this type. We begin with the power-law convection-diffusion equation, proceed to the power-law reaction-diffusion equation with linear convection, and, end with the full equation.

Brian H. Gilding, Robert Kersner

Chapter 13. Explicit travelling-wave solutions

Abstract

A number of explicit nontrivial monotonic travelling-wave solutions of the nonlinear reaction-convection-diffusion equation (1.1) have been discovered by various authors. It is not the intention here to provide a survey of all these. However, a few remarks on the possibilities offered by the now apparent correspondence between travelling-wave solutions of the partial differential equation (1.1) and solutions of the integral equation (1.9) are in order.

Brian H. Gilding, Robert Kersner

Backmatter

Titel: Travelling Waves in Nonlinear Diffusion-Convection Reaction
herausgegeben von: Brian H. Gilding
Robert Kersner
Verlag: Birkhäuser Basel
Electronic ISBN: 978-3-0348-7964-4
Print ISBN: 978-3-0348-9638-2
DOI: https://doi.org/10.1007/978-3-0348-7964-4

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Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction

Chapter 2. General theory

Chapter 3. Transformations

Chapter 4. Travelling waves

Chapter 5. Convection-diffusion

Chapter 6. Reaction-diffusion

Chapter 7. Power-law equations

Chapter 8. Wavefronts

Chapter 9. Wavefronts for convection-diffusion

Chapter 10. Wavefronts for reaction-diffusion

Chapter 11. Unbounded waves

Chapter 12. Wavefronts and unbounded waves for power-law equations

Chapter 13. Explicit travelling-wave solutions

Backmatter