Effect of a cutoff on pushed and bistable fronts of the reaction-diffusion equation

Abstract

We give an explicit formula for the change of speed of pushed and bistable fronts of the reaction diffusion equation when a small cutoff is applied at the unstable or metastable equilibrium point.The results are valid for arbitrary reaction terms and include the case of density dependent diffusion.

A geometric approach to bistable front propagation in scalar reaction–diffusion equations with cut-off

“…In addition to proving the existence of traveling front solutions, we will calculate explicitly the ε-dependent correction ∆c(ε) to the front propagation speed c 0 that is induced by the cut-off in (1.6). In particular, we will prove that this correction is positive, i.e., that the propagation speed of bistable fronts increases in the presence of a cut-off, which is in agreement with results reported previously in [5,7]. Moreover, we will show that ∆c scales with fractional powers of the cut-off parameter ε, and we will provide explicit expressions for these exponents, as well as -in certain cases -for the respective leading-order coefficients in the expansion for ∆c(ε).…”

“…(9)], which implies ∆c ∼ −Kf (0)ε 1+λ , is equivalent to (2.2), with f (0) = −γ and λ = 2γ . However, the numerical value of K γ , as stated in (2.3), differs from that reported for K in [5] by a multiplicative factor of (1 + 2γ ) −2γ . While the reason for this discrepancy warrants further investigation, we are confident that (2.3) is correct.…”

“…(As shown e.g. in [5,6], the front speed c 0 is unique in the bistable case, whereas the FKPP equation supports traveling front solutions for a continuum of speeds c in the absence of a cut-off, with c ≥ c FKPP (0) and c FKPP (0) the 'critical' front speed; cf. [3] for a detailed discussion.)…”

“…The traveling front equation corresponding to (2.1) may be expressed as the equivalent firstorder system [7] and in [5]. In particular, [5,Eq.…”

“…The following corollary is the main result of this section: Proof. The result is most easily seen from the well-known equivalence of the Schlögl and Nagumo equations [5,7]: introducing the new variables…”