Stability of a front for a nonlocal conservation law

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.