Reaction-diffusion waves in an isothermal chemical system with general orders of autocatalysis and spatial dimension

Merkin, J. H.; Needham, D. J.

doi:10.1007/bf00948484

Cited by 29 publications

(54 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This contrasts with the fact that the critical wave speed c FKPP = 2 in the continuum limit (1) is determined locally, see [2,4,8,20,24,29]. In the corresponding travelling wave ODE, the origin is a stable node for c > 2, a degenerate stable node at c = 2 and a stable spiral when c < 2.…”

Section: Theorem 11 For Any Reaction-diffusion Equation Of the Formmentioning

confidence: 96%

The critical wave speed for the Fisher–Kolmogorov–Petrowskii–Piscounov equation with cut-off

2007

View full text Add to dashboard Cite

The Fisher-Kolmogorov-Petrowskii-Piscounov (FKPP) equation with cut-off was introduced in (Brunet and Derrida 1997 Shift in the velocity of a front due to a cut-off Phys. Rev. E 56 2597-604) to model N-particle systems in which concentrations less than ε = 1/N are not attainable. It was conjectured that the cut-off function, which sets the reaction terms to zero if the concentration is below the small threshold ε, introduces a substantial shift in the propagation speed of the corresponding travelling waves. In this paper, we prove the conjecture of Brunet and Derrida, showing that the speed of propagation is given by c crit (ε) = 2 − π 2 /(ln ε) 2 + O((ln ε) −3 ), as ε → 0, for a large class of cut-off functions. Moreover, we extend this result to a more general family of scalar reaction-diffusion equations with cut-off. The main mathematical techniques used in our proof are the geometric singular perturbation theory and the blow-up method, which lead naturally to the identification of the reasons for the logarithmic dependence of c crit on ε as well as for the universality of the corresponding leading-order coefficient (π 2 ).

show abstract

Section: Theorem 11 For Any Reaction-diffusion Equation Of the Formmentioning

confidence: 96%

The critical wave speed for the Fisher–Kolmogorov–Petrowskii–Piscounov equation with cut-off

2007

View full text Add to dashboard Cite

show abstract

“…In both cases, (S) is uniformly globally well-posed on U 0+ . However, for p, q ≥ 1, u(x, t) → 1 as t → ∞ through the propagation of finite speed travelling wave structures [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16], whereas, for 0 < p, q < 1, u(x, t) → 1 uniformly for x ∈ R (through uniform terms of O(t −1/2 ) as t → ∞), as demonstrated in this paper. In fact, we can now immediately infer stability…”

Section: Theorem 52 (Continuous Dependence)mentioning

confidence: 66%

“…In particular, classical Hadamard well-posedness has been established, along with considerable qualitative information regarding the structure of the solution to (1.1)-(1.3). Specific attention has been focused on the convergence to the equilibrium state u = 1 via the evolution of travelling wave structures in the solution to (1.1)-(1.5) when the initial data is non-trivial, as t → ∞, their propagation speed, shape and form [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16]. The cases when 0 < p < 1 and/or 0 < q < 1 have received much less attention, primarily because the nonlinearity f : R → R lacks Lipschitz continuity in these cases owing to the behaviour at u = 0 and/or u = 1 and the classical comparison theorems and continuous dependence theorems fail to apply.…”

Section: Introductionmentioning

confidence: 99%

Well-posedness and qualitative behaviour of a semi-linear parabolic Cauchy problem arising from a generic model for fractional-order autocatalysis

Meyer

Needham

2015

Proc. R. Soc. A.

Self Cite

View full text Add to dashboard Cite

“…This will lead to the concentration gradients, and then the advancing wave front of the reactant A (or the autocatalyst B) is thus generated and will propagate out from this initial zone. Mathematically, Billingham, Merkin, and Needham [25,5,6,28] have applied the numerical and asymptotical analysis to the system (1.1) with f (u) = k 1 u and K = 0 to confirm such a phenomenon of wave front propagation. We note that the rigorous proof of the existence of traveling wave front solutions to the system (1.1) with f (u) = k 1 u and K = 0 is given by Qi [30] and Chen and Qi [8] (see also Marion [24] and Ai and Huang [2] for the general reaction term f ).…”

mentioning

confidence: 99%

“…On the other hand, the analysis by Billingham, Merkin, and Needham [25,5,6,28] shows that no matter how small the amount of the autocatalyst B is introduced locally into the system (1.1) with f (u) = k 1 u and K = 0, traveling waves are always generated. This particular feature seems to be contrary to the fact that in many chemical systems, the initial input of an autocatalyst into the system must be above a threshold concentration for the initialization of traveling waves.…”

mentioning

confidence: 99%

Existence of traveling waves in a simple isothermal chemical system with the same order for autocatalysis and decay

Tsai

2010

Quart. Appl. Math.

View full text Add to dashboard Cite

Abstract. The reaction-diffusion systems which are based on an isothermal autocatalytic chemical reaction involving both an autocatalytic step of the (m + 1)th order (A + mB → (m + 1)B) and a decay step of the same order (B → C) have very rich and interesting dynamics. Previous studies in the literature indicate that traveling waves play a key role in understanding these interesting dynamical phenomena. However, there is a lack of rigorous proof of the existence of traveling waves to this system. Here we generalize this isothermal autocatalytic chemical reaction model and provide a rigorous proof of the existence of traveling waves for the resulting reaction-diffusion system which also includes the systems arising from epidemiology and the microbial growth in a flow reactor. Introduction.Since the pioneering works of Fisher [9] and of Kolmogorov, Petrovsky and Piskunov [23], traveling waves in a wide array of biological and chemical systems, based on the interaction between reaction and diffusion processes, have been extensively investigated. In many realistic systems, traveling waves can be initiated by a highly localized disturbance of a stable rest state (e.g., a highly localized increase in ion concentrations). A typical example is the Hodgkin-Huxley model [13] which describes the electrical activity across membranes of never cells. It is known that if a short, but sufficiently large, burst of simulating current is injected into a nerve axon, a so-called action potential wave can then be generated. In this paper, we will discuss a system which enjoys a similar phenomenon. Specifically, this system is governed by the following equations:

show abstract

Reaction-diffusion waves in an isothermal chemical system with general orders of autocatalysis and spatial dimension

Cited by 29 publications

References 14 publications

The critical wave speed for the Fisher–Kolmogorov–Petrowskii–Piscounov equation with cut-off

The critical wave speed for the Fisher–Kolmogorov–Petrowskii–Piscounov equation with cut-off

Well-posedness and qualitative behaviour of a semi-linear parabolic Cauchy problem arising from a generic model for fractional-order autocatalysis

Existence of traveling waves in a simple isothermal chemical system with the same order for autocatalysis and decay

Contact Info

Product

Resources

About