On the nonlinear diffusion equation of Kolmogorov-Petrovskii-Piskunov type

Abstract

Page Introduction Part I. Travelling waves 1. Existence, uniqueness and properties of the travelling wave 2. KPP transform of the travelling wave 3. Second existence theorem for the travelling wave Part II. Asymptotic behavior of the time dependent solution 4. Stability and instability criteria for ιυ\(x) 5. A problem with higher space dimension 6. Second stability theorem Part III. Method of KPP transform 7. Fundamental theorem of KPP 8. Stability of the slowest travelling wave 9. Stability of the travelling wave with arbitrary speed 10. An example References 66 /^.oo XtζA or lim sup \u(x 9 t)-zϋ λ (x-{-c)\ = 0 for some c.

Further probabilistic analysis of the Fisher–Kolmogorov–Petrovskii–Piscounov equation: one sided travelling-waves

“…Concerning existence, see also Watanabe [37]. Theorem 1 can be also be extracted from Kametaka [23] who also uses classical phase-plane analysis techniques (as in Coddington and Levinson [11]), although some care is required as this paper is predominantly concerned with the opposing case of ρ ≥ √ 2β. In the spirit of Harris [18] and Kyprianou [27], we shall devote the first part of this article to a new proof of Theorem 1 using probabilistic means alone which, for the most part, means that we appeal either to martingale arguments, 'spine' decompositions, or fundamental properties of both branching and single-particle Brownian motion.…”

“…From the definition for W λ , it is clear that {ζ < ∞} ⊆ {W λ (∞) = 0}, so that P x (W λ (∞) = 0; ζ < ∞) = P x (ζ < ∞). We can also write P x (W λ (∞) = 0) as (23) and the the result follows if we can show that P x (W λ (∞) = 0) = P x (ζ < ∞). Define g(x) := P x (W λ (∞) = 0), and then, by a similar argument to that used in the proof of Theorem 13…”

Asymptotic behavior and traveling wave solutions for parabolic functional-differential equations

“…For equations without delay, criterion 2.1O(i) can be found in [19]. One can show the continuous dependence of the wave solutions on c and r in the sense of Theorem 3.16; the proof is omitted here.…”

Existence of Positive Solutions of Fisher-KPP Equations in Locally Spatially Variational Habitat with Hybrid Dispersal

“…We refer to (Aronson & Weinberger, 1957;Aronson & Weinberger, 1978;Berestycki, Hamel, & Nadirashvili, 2010;Kametaka, 1976;Liang & Zhao, 2007;Liang, Yi, & Zhao, 2006;Sattinger, 1976;Uchiyama, 1978;Weinberger, 1982;etc). for the study of (1.2) in the case that f (x, u) is independent of x and refer to (Berestycki, Hamel, & Nadirashvili, 2005;Berestycki, Hamel, & Roques, 2005;Freidlin & Gärtner, 1979;Hamel, 2008;Hudson & Zinner, 1995;Nadin, 2009;Nolen, Rudd, & Xin, 2005;Weinberger, 2002;etc).…”