Travelling wave behaviour for a Porous-Fisher equation

Abstract

We study the propagation properties of the reaction-diffusion equation of Fisher type u t l (u m−" u x) x ju p (1ku) for x ? , t 0, with p 1 m. Taking into account that solutions of the Cauchy problem are nonunique if mjp 2, we prove that the minimal solutions in this case tend to propagate with certain minimal speed cJ(m, p). More precisely, if we translate any solution with a velocity QcQ cJ, we get the limit in time value one, and if the initial value u(:, 0) vanishes, say, for x 0, the minimal solution translated with velocity c cJ tends to zero. Also, an interface appears for the minimal solutions, whose asymptotic velocity is cJ. This behaviour depends upon the existence of special solutions of travelling wave form. Travelling waves have been widely studied for diffusion equations related with the above. We characterize here the minimal velocity cJ for which travelling waves exist, as an analytic function of the parameters m and p, for every mjp 2, by viewing it as an anomalous exponent. Some local properties of the minimal solutions and their interfaces in the case mjp 2 are also proved.

Existence and Stability of Travelling Front Solutions for General Auto-catalytic Chemical Reaction Systems

“…It has been proved in [6] and [26] that there exists a minimal speed c 0 * (p) > 0 such that there exists a travelling wave V c (z) of (1.5)-(1.6) if and only if c ≥ c 0 * (p), and V c (z) satisfies V ′ c (z) < 0 for z ∈ R and * (p) were investigated in [22] and [10]. By applying similar phase plane analysis as in [6] and applying center manifold theorems, the above mentioned existence results for p degree Fisher equations are still valid for equation (1.5) with more general f (v) satisfying f ∈ C 2 ((0, 1]), f (0) = 0, f (v) > 0 for v ∈ (0, 1], and lim v→0 + f (v) v p = k 0 > 0 for some p > 1, (1.10) i.e.…”

Travelling Waves in Nonlinear Diffusion-Convection Reaction

“…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.…”

“…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].…”

“…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.…”

“…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.…”