Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-26T17:09:29.201Z Has data issue: false hasContentIssue false
  • English
  • Français

Stability of a front for a nonlocal conservation law

Published online by Cambridge University Press:  14 November 2011

Amine Asselah
Amine Asselah
Affiliation:
Department of Mathematics, Rutgers University, Hill Center, Busch Campus, New Brunswick NJ 08903, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the stability of a front for the law 2wt − (wx − γ(1 − w2)(K * w)x)x = 0. It was proved by Del Passo and De Mottoni that an increasing stationary solution, u, exists. We show that it is stable in the following sense: there is ε > 0 such that if w(0) = u + v with |v|2 < ε, then there is α(t) differentiable such that w(x, t) = u(α(t) + x) + v(x, t) and supℝ |v(x, t)| converges to 0 as t goes to infinity. Also, if v is initially odd, α(t) ≡ 0.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 128 , Issue 2 , 1998 , pp. 219 - 234
Copyright
Copyright © Royal Society of Edinburgh 1998

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Passo, R. Del and Mottoni, P. De. The heat equation with a nonlocal density dependent advection term (Preprint, 1991).Google Scholar
2De Masi, A., Gobron, T. and Presutti, E.. Travelling fronts in nonlocal evolution equations. Arch. Rational Mech. Anal. 132 (1995), 143–205.CrossRefGoogle Scholar
3De Masi, A., Orlandi, E., Presutti, E. and Triolo, L.. Stability of the interface in a model of phase separation. Proc. Roy. Soc. Edinburgh Sect. A 124 (1994), 1013–22.CrossRefGoogle Scholar
4De Masi, A., Orlandi, E., Presutti, E. and Triolo, L.. Uniqueness and global stability of the instanton in non local evolution equations. Rend. Mat. Appl. (7) 14 (1994), 693–723.Google Scholar
5Lebowitz, J. L., Orlandi, E. and Presutti, E.. A particle model for spinodal decomposition. J. Stat. Phys. 63 (1991), 933–74.CrossRefGoogle Scholar