## Differential and Integral Equations

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

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

#### 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 $\varepsilon$.

#### Article information

**Source**

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

**Dates**

First available in Project Euclid: 20 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1356039441

**Mathematical Reviews number (MathSciNet)**

MR2324218

**Zentralblatt MATH identifier**

1212.35294

**Subjects**

Primary: 35L60: Nonlinear first-order hyperbolic equations

Secondary: 35B25: Singular perturbations 35K55: Nonlinear parabolic equations 37L99: None of the above, but in this section

#### Citation

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