Abstract
We study the existence of monotone traveling waves u(t, x )= u(x + ct), connecting two equilibria, for the reaction-diffusion PDE ut =( ux √ 1+u 2 x )x + f (u). Assuming different forms for the reaction term f (u) (among which we have the so-called types A, B, and C), we show that, concerning the admissible speeds, the situation presents both similarities and differences with respect to the classical case. We use a first order model obtained after a suitable change of variables. The model contains a singularity and therefore has some features which are not present in the case of linear diffusion. The technique used involves essentially shooting arguments and lower and upper solutions. Some numerical simulations are provided in order to better understand the features of the model. MSC: 34C37; 35K57; 34B18
Highlights
In this paper, we will be interested in the existence of monotone heteroclinic solutions for a quasilinear variant of the scalar second order differential equation u – cu + f (u) =, ( )which arises in connection with the Fisher-Kolmogorov PDE ut = uxx + f (u) when searching for traveling waves with speed c, i.e., solutions having the form u(t, x) = u(x + ct)
We study the existence of monotone traveling waves u(t, x) = u(x + ct), connecting two equilibria, for the reaction-diffusion
Assuming different forms for the reaction term f (u), we show that, concerning the admissible speeds, the situation presents both similarities and differences with respect to the classical case
Summary
Turning to more general types of reaction, let us further observe that it suffices that there exists < u ≤ u such that f (u) < for every u ∈ ] , u [, with f (u ) = (i.e., f is negative in a neighborhood of ) for the admissible speed (if any) to be unique, as it is possible to see with the usual monotonicity argument applied focusing on y+c,f (u ), y–c,f (u ). In this case it is not sufficient that ( ) and ( ) hold in order to have existence of an admissible speed, as the following example shows.
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