dorsal/arxiv
View SchemaExistence and Stability of Propagating Fronts for an Autocatalytic Reaction-Diffusion System
| Authors | Stephane Focant, Thierry Gallay |
|---|---|
| Categories | |
| ArXiv ID | patt-sol/9705008 |
| URL | https://arxiv.org/abs/patt-sol/9705008 |
| DOI | 10.1016/S0167-2789(98)00096-7 |
Abstract
We study a one-dimensional reaction-diffusion system which describes an isothermal autocatalytic chemical reaction involving both a quadratic (A + B -> 2B) and a cubic (A + 2B -> 3B) autocatalysis. The parameters of this system are the ratio D = D_B/D_A of the diffusion constants of the reactant A and the autocatalyst B, and the relative activity k of the cubic reaction. First, for all values of D > 0 and k >= 0, we prove the existence of a family of propagating fronts (or travelling waves) describing the advance of the reaction. In particular, in the quadratic case k=0, we recover the results of Billingham and Needham [BN]. Then, if D is close to 1 and k is sufficiently small, we prove using energy functionals that these propagating fronts are stable against small perturbations in exponentially weighted Sobolev spaces. This extends to our system part of the stability results which are known for the scalar Fisher equation.
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"abstract": "We study a one-dimensional reaction-diffusion system which describes an\nisothermal autocatalytic chemical reaction involving both a quadratic (A + B -\u003e\n2B) and a cubic (A + 2B -\u003e 3B) autocatalysis. The parameters of this system are\nthe ratio D = D_B/D_A of the diffusion constants of the reactant A and the\nautocatalyst B, and the relative activity k of the cubic reaction. First, for\nall values of D \u003e 0 and k \u003e= 0, we prove the existence of a family of\npropagating fronts (or travelling waves) describing the advance of the\nreaction. In particular, in the quadratic case k=0, we recover the results of\nBillingham and Needham [BN]. Then, if D is close to 1 and k is sufficiently\nsmall, we prove using energy functionals that these propagating fronts are\nstable against small perturbations in exponentially weighted Sobolev spaces.\nThis extends to our system part of the stability results which are known for\nthe scalar Fisher equation.",
"arxiv_id": "patt-sol/9705008",
"authors": [
"Stephane Focant",
"Thierry Gallay"
],
"categories": [
"patt-sol",
"nlin.PS"
],
"doi": "10.1016/S0167-2789(98)00096-7",
"title": "Existence and Stability of Propagating Fronts for an Autocatalytic Reaction-Diffusion System",
"url": "https://arxiv.org/abs/patt-sol/9705008"
},
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