A geometric approach to bistable front propagation in scalar reaction-diffusion equations with cut-off
This paper is built on the results obtained by the same authors during 2007. The cut-offs problems are introduced to model front propagation in reaction-diffusion systems in which the reaction is effectively deactivated at points where the concentration lies below some threshold. This paper is devoted to the investigation of effects of a cut-off on fronts propagating into metastable states in a class of bistable scalar equations. The main problem considered here is the parabolic equation with cut-off of the form \[ \varphi_t=\varphi_{xx}+ f(\varphi) \Theta\bigg(\varphi,\varepsilon,\frac{\varphi}{\varepsilon}\bigg), \] where \((t,x)\in\mathbb R^+\times\mathbb R\), \(\varphi (t,x)\in\mathbb R\), and \(f:\mathbb R\to\mathbb R\) denotes a smooth reaction function which vanishes at the three rest states at \(\varphi^+\), \(\varphi^o\), \(\varphi^-\) in the above stated equation. The cut-off function satisfies \[ \Theta\bigg(\varphi,\varepsilon,\frac{\varphi}{\varepsilon}\bigg)<1, \quad \text{if }\varphi<\varepsilon, \] \[ \Theta\bigg(\varphi,\varepsilon,\frac{\varphi}{\varepsilon}\bigg)\equiv 1, \quad \text{if }\varphi>\varepsilon, \] where \(\Theta \) is bounded at \(\varphi=\varepsilon \) and \(0<\varepsilon\ll1\) denotes cut-off parameter. The method of geometric desingularization from dynamical systems theory is applied here in order to calculate explicitly the change in front propagation speed that is induced by the cut-off. It is shown that the speed of bistable fronts increases in the presence of a cut-off, in agreement with results obtained previously via a variational principle. The classical Nagumo equation is discussed as a prototypical example of bistable front propagation. Corresponding results for Schlögl equation with cut-off \[ \varphi_t=\varphi_{xx}- [2(1-\sigma)\varphi+ (\sigma-3)\varphi^2+ \varphi^3]H(\varphi- \varepsilon), \quad \sigma\in (0,1), \] where \(H\) is the Heaviside cut-off, are considered as well. An interesting conclusion is that the knowledge of an explicit front solution to the associated problem without cut-off is necessary for the correction induced by the cut-off to be computable in closed form.
scientific article; zbMATH DE number 5810798
scientific article