article

Title: Uniform convergence for approximate traveling waves in linear reaction-hyperbolic systems

Authors: Avner Friedman and Bei Hu

Issue: Volume 56 (2007), Issue 5, 2133-2158

Abstract:

$In this paper we study a linear reaction-hyperbolic systems of the form $\epsilon (\partial_t + v_i \partial_x) p_i = \sum_{j=1}^n k_{ij} p_j$ ($i = 1, 2, \dots, n$) for $x > 0$, $t > 0$ with "near equilibrium" initial and boundary data. This problem arises in a model of transport of neurofilaments in axons. The matrix $(k_{ij})$ is assumed to have a unique null vector $(\lambda_1, \dots, \lambda_n)$ with positive components summed to $1$ and the $v_j$ are arbitrary velocities such that $v \equiv sum_{j=1}^n \lambda_j v_j > 0$. We prove that as $\epsilon \to 0$, the solution converges to a traveling wave with velocity $v$ and a spreading front, and that the convergence rate in the uniform norm is $O(\epsilon^{(1 - \alpha)/2})$, for any small positive $\alpha$.$