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

In this paper we study linear reaction–hyperbolic systems of the form , (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 with positive components summed to 1 and the v _j are arbitrary velocities such that . We prove that as the solution converges to a traveling wave with velocity v and a spreading front, and that the convergence rate in the uniform norm is , for any small positive α.     

On Strongly Interacting Internal Solitary Waves

The Cauchy problem and global well-posedness for a mathematical model of the strong interaction of two-dimensional, long, internal gravity waves propogating on neighboring pycnoclines in a stratified fluid have been studied by Bona, Ponce, Saut, Tom, and others. We show that global well-posedness occurs even when the initial data is rough.