Differential and Integral Equations

Front speeds in the vanishing diffusion limit for reaction-diffusion-convection equations

E. C. M. Crooks and C. Mascia

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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 $\varepsilon$.

Article information

Differential Integral Equations, Volume 20, Number 5 (2007), 499-514.

First available in Project Euclid: 20 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35L60: Nonlinear first-order hyperbolic equations
Secondary: 35B25: Singular perturbations 35K55: Nonlinear parabolic equations 37L99: None of the above, but in this section


Crooks, E. C. M.; Mascia, C. Front speeds in the vanishing diffusion limit for reaction-diffusion-convection equations. Differential Integral Equations 20 (2007), no. 5, 499--514. https://projecteuclid.org/euclid.die/1356039441

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