Uniform Convergence for Approximate Traveling Waves in Linear Reaction–Diffusion–Hyperbolic Systems

Abstract

In this paper we study linear reaction–hyperbolic systems of the form $\varepsilon (\partial_t + v_i \partial_x) p_i = \sum_{j=0}^n k_{ij} p_j$ , (i =  1, 2, ..., n) for x >  0, t >  0 coupled to a diffusion equation for p ₀ =  p ₀(x, y, θ, t) 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 $(\lambda_0,\lambda_1,\ldots,\lambda_n)$ with positive components summed to 1 and the v _j are arbitrary velocities such that $v \equiv \frac{1}{1-\lambda_0} \sum_{j=1}^n \lambda_j v_j > 0$ . We prove that as $\varepsilon \to 0$ the solution converges to a traveling wave with velocity v and a spreading front, and that the convergence rate in the uniform norm is $O(\varepsilon^{(1-\alpha)/2})$ , for any small positive α.