Travelling waves and finite propagation in a reaction-diffusion equation

Pablo, Arturo de; Vázquez, Juan Luís

doi:10.1016/0022-0396(91)90021-z

Cited by 132 publications

(116 citation statements)

References 13 publications

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“…Rather than presenting the most general result possible, we present a conclusion which succinctly covers the previous work [15,22,124,207,209,232]. We use the next two lemmas.…”

Section: Wavefront Propertiesmentioning

confidence: 90%

“…Since the pioneering work of Fisher [98] and of Kolmogorov, Petrovskii and Piskunov [172], much attention has been paid to the study of wavefront solutions of reaction-diffusion equations of the class (10.1) [25,53,78,93,123,191,192,267,268] [207], de Pablo and Vázquez [209], Pauwelussen and Peletier [213], Sánchez-Garduño and Maini [232], Uchiyama [254,256], and, Vol'pert, Vol'pert and Vol'pert [268]. Many of these results have also been extended to a higher number of dimensions in [42,43].…”

Section: Wavefronts For Reaction-diffusionmentioning

confidence: 99%

“…These questions have received considerable interest in recent years, significant contributions having been made in chronological order by Aronson [15], Atkinson, Reuter and Ridler-Rowe [22], Grindod and Sleeman [124], de Pablo and Vázquez [209], Sánchez-Garduño and Maini [232], and, de Pablo and Sánchez [207]. Terms which have been used to designate wavefront solutions satisfying (10.40), include "weak" (sic) [124], finite [18,148,207,209], and, of sharp type [232]. Oppositely to the last two terms, wavefront solutions for which (10.41) holds have been referred to as positive [207], and, of front type [232], respectively.…”

Section: Wavefront Propertiesmentioning

confidence: 99%

“…The first is Murray's model of biological dispersal in which the unknown u represents population density [192]. This is the equation [15] and by Atkinson, Reuter and Ridler-Rowe [22], for m > 1 and q = 1 by de Pablo and Vázquez [209], for m > 1, p < 1 and q = 1 by de Pablo and Sánchez [207], for m > 1 and p ≥ 1 by Biro [45,46], and, for m = 1, p ≥ 1 and q = 1 by Needham and Barnes [196]. 50) and,…”

Section: Proof Note That For This Examplementioning

confidence: 99%

“…have been studied in great depth by Herrero and Vázquez [148] and by de Pablo and Vázquez [209]. In the earlier study [148], equation (12.7) was analysed with c 0 < 0 and no restriction on m and p. In the later paper, the subject was (12.7) with c 0 > 0, m > 1 and no restriction on p. In both of these papers travelling waves were studied by means of a detailed phase-plane analysis.…”

Section: Reaction-diffusion With Linear Convectionmentioning

confidence: 99%

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Travelling Waves in Nonlinear Diffusion-Convection Reaction

Gilding¹,

Kersner²

2004

158

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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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“…Rather than presenting the most general result possible, we present a conclusion which succinctly covers the previous work [15,22,124,207,209,232]. We use the next two lemmas.…”

Section: Wavefront Propertiesmentioning

confidence: 90%

Section: Wavefronts For Reaction-diffusionmentioning

confidence: 99%

Section: Wavefront Propertiesmentioning

confidence: 99%

Section: Proof Note That For This Examplementioning

confidence: 99%

Section: Reaction-diffusion With Linear Convectionmentioning

confidence: 99%

See 3 more Smart Citations

Travelling Waves in Nonlinear Diffusion-Convection Reaction

Gilding¹,

Kersner²

2004

158

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Traveling waves for generalized Fisher–Kolmogorov equation with discontinuous density dependent diffusion

Drábek

Zahradníková

2022

Math Methods in App Sciences

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We are concerned with the existence and qualitative properties of traveling wave solutions for a quasilinear reaction-diffusion equation on the real line. We consider a non-Lipschitz reaction term of Fisher-KPP type and a discontinuous diffusion coefficient that allows for degenerations and singularities at equilibrium points. We investigate the joint influence of the reaction and diffusion terms on the existence and nonexistence of traveling waves, and assuming these terms are of power-type near equilibria, we provide classification of solutions based on their asymptotic properties. Our approach provides a broad theoretical background for the mathematical treatment of rather general models not only in population dynamics but also in other applied sciences and engineering.

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Regularity and profiles of solutions to a higher order Eyring–Powell fluid with Darcy–Forchheimer porosity term

Rahman

Palencia

2022

Math Methods in App Sciences

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A flow of Eyring–Powell type constitutes a remarkable area of analysis to model non‐Newtonian processes in fluids. The associated diffusion term comes from the general kinetic theory of liquids and permits to account for a wider diffusivity, which is applicable for qualitatively low to higher shear stresses. The goal of the present article is to introduce a generalization of an Eyring–Powell fluid by the introduction of a porous reaction term (of Darcy–Forchheimer type) and a perturbation with a higher order operator. In particular, we consider that our model is an extension of a classical Eyring–Powell fluid in the same manner as introduced for other equations (see the extended Fisher–Kolmogorov model). The obtained equation is novel and requires analysis about existence, regularity and uniqueness of solutions. Stationary solutions are explored under the definition of a Hamiltonian. In addition, profiles of solutions are obtained with an exponential scaling that ends in a Hamilton–Jacobi equation. Eventually, some numerical assessments are introduced to validate the hypothesis done, and to discuss about the accuracy of the analytical approach followed.

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Travelling waves and finite propagation in a reaction-diffusion equation

Cited by 132 publications

References 13 publications

Travelling Waves in Nonlinear Diffusion-Convection Reaction

Travelling Waves in Nonlinear Diffusion-Convection Reaction

Traveling waves for generalized Fisher–Kolmogorov equation with discontinuous density dependent diffusion

Regularity and profiles of solutions to a higher order Eyring–Powell fluid with Darcy–Forchheimer porosity term

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