Differential and Integral Equations

Front profiles in the vanishing-diffusion limit for monostable reaction-diffusion-convection equations

E.C.M. Crooks

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We prove the vanishing-viscosity $L^1({\mathbb{R}})$-convergence of minimal-speed travelling-front profiles for scalar balance laws with monostable reaction, possibly non-convex flux, and viscosity $\varepsilon \geq 0$. Such equations are known to admit so-called entropy travelling fronts for all velocities greater than or equal to an $\varepsilon$-dependent minimal value, both for $\varepsilon >0$, when all fronts are smooth, and for $\varepsilon =0$, when the possibly non-convex flux results in fronts of speed close to the minimal value typically having discontinuities where jump conditions hold. Our main result is that, as $\varepsilon \downarrow 0$, profiles of minimal velocity ${c_{\varepsilon}^*}$ converge in $L^1({\mathbb{R}})$ to the unique (up-to-translation) $\varepsilon=0$ entropy-front profile of minimal velocity ${c^*}$. The proofs exploit the compactness inherent in the monotonicity of the profiles, together with uniform-in-$\varepsilon$ estimates on the convergence of the profiles to their spatial limits. Convergence results for the less-delicate fronts of non-minimal speed also follow from the arguments given.

Article information

Differential Integral Equations, Volume 23, Number 5/6 (2010), 495-512.

First available in Project Euclid: 20 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35B25: Singular perturbations 35K57: Reaction-diffusion equations 35L65: Conservation laws 34C37: Homoclinic and heteroclinic solutions


Crooks, E.C.M. Front profiles in the vanishing-diffusion limit for monostable reaction-diffusion-convection equations. Differential Integral Equations 23 (2010), no. 5/6, 495--512. https://projecteuclid.org/euclid.die/1356019308

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