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2007 Front speeds in the vanishing diffusion limit for reaction-diffusion-convection equations
E. C. M. Crooks, C. Mascia
Differential Integral Equations 20(5): 499-514 (2007).

Abstract

Travelling fronts for scalar balance laws with monostable reaction, possibly non-convex flux, and viscosity $\varepsilon \geq 0$ exist for all velocities greater than or equal to an $\varepsilon$-dependent minimal value, both in the parabolic case when $\varepsilon >0$ and in the hyperbolic case when $\varepsilon =0$. We prove that as $\varepsilon \rightarrow 0$, the minimal velocity ${c_{\varepsilon}^*}$ converges to $c^*$, the minimal value when $\varepsilon =0$, and that ${c_{\varepsilon}^*}\geq c^*$ for all $\varepsilon >0$. The proof uses comparison theorems and the variational characterization of the minimal parabolic front velocity. This convergence also yields a reaction-independent sufficient condition for the minimal velocity of the parabolic problem for small positive $\varepsilon$ to be strictly greater than the value predicted by the problem linearized about the unstable equilibrium, that is, for the minimal-velocity travelling front of the viscous equation to be pushed for sufficiently small $\varepsilon$.

Citation

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E. C. M. Crooks. C. Mascia. "Front speeds in the vanishing diffusion limit for reaction-diffusion-convection equations." Differential Integral Equations 20 (5) 499 - 514, 2007.

Information

Published: 2007
First available in Project Euclid: 20 December 2012

zbMATH: 1212.35294
MathSciNet: MR2324218

Subjects:
Primary: 35L60
Secondary: 35B25 , 35K55 , 37L99

Rights: Copyright © 2007 Khayyam Publishing, Inc.

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Vol.20 • No. 5 • 2007
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