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

In this paper we study a linear reaction-hyperbolic systems of the form epsilon(partial derivative(t) + v(i)partial derivative(x))p(i) = Sigma(n)(j=1) k(ij)p(j) (i = 1, 2, ..., 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), ..., lambda(n)) with positive components summed to 1 and the v(j) are arbitrary velocities such that v equivalent to Sigma(n)(j=1) lambda(j)v(j) > 0. We prove that as epsilon -> 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.