A geometric analysis of front propagation in a family of degenerate reaction-diffusion equations with cutoff

Abstract

We investigate the effects of a Heaviside cutoff on the dynamics of traveling fronts in a family of scalar reaction-diffusion equations with degenerate polynomial potential that includes the classical Zeldovich equation. We prove the existence and uniqueness of front solutions in the presence of the cutoff, and we derive the leading-order asymptotics of the corresponding propagation speed in terms of the cutoff parameter. For the Zeldovich equation, an explicit solution to the equation without cutoff is known, which allows us to calculate higher-order terms in the resulting expansion for the front speed; in particular, we prove the occurrence of logarithmic (switchback) terms in that case. Our analysis relies on geometric methods from dynamical systems theory and, in particular, on the desingularization technique known as 'blow-up.' (2000). Primary 35K57 · 34E15; Secondary 34E05. Mathematics Subject Classification

A geometric analysis of front propagation in an integrable Nagumo equation with a linear cut-off

“…Following [3,4,10], we consider front propagation in the cut-off Nagumo equation in the framework of the equivalent first-order system that is obtained by introducing u ′ = v in (6). Appending the trivial ε-dynamics to the resulting equations, we find…”

“…While the propagation speed of pulled fronts is selected by linearization about that state, the corresponding selection mechanism in the pushed and bistable regimes is highly nonlinear; see e.g. [2,3,4] for details and references. In particular, the bistable regime in (1) is realized for γ ∈ (0, 1 2 ), in which case c 0 is positive; at the so-called Maxwell point, which is defined by γ = 1 2 , the speed vanishes, and the front solution in (3) corresponds to a stationary (time-independent) solution of Equation 1, as ξ = x then.…”

“…Geometric desingularization has since been successfully applied in the study of the effects of a cut-off on propagating fronts in the Nagumo equation: the bistable regime, with γ ∈ (0, 1 2 ) in (1), was analyzed in [3], while the 'boundary' case where γ = 0 was discussed in detail in [4]. (Here, we remark that Equation 1is also known as the Zeldovich equation in that case [16].)…”

Geometric desingularization of degenerate singularities in the presence of fast rotation: A new proof of known results for slow passage through Hopf bifurcations

“…The geometric desingularization method [9], also known as the blow-up method, has been widely used in the analysis of fast-slow systems of differential equations with nilpotent singularities. It is central to the study of limit cycle canards [15,16,28,30,31,52,53], canards and dynamics near singularities and folded singularities [7,32,54,56,57], certain defects and patterns [44], traveling wave solutions of a variety of reaction-diffusion equations [3,11,12,13,14,25,26,40,41,42], and a series of other problems [17,19,22,34,39,43,45,46,47,48,58].…”

Shift in the speed of reaction–diffusion equation with a cut-off: Pushed and bistable fronts

“…When a = 0, f ′ (0) = 0 and the shift is obtained using (15). In this case f (ǫ) = ǫ 2 to leading order, c 0 = 1/ √ 2 and g 0 (u) = (1 − u)/u so that…”