Traveling wave to a reaction-hyperbolic system for axonal transport

In this paper, we study a class of nonlinear reaction-hyperbolic systems modeling the neuronal signal transfer in neuroscience. This reaction-hyperbolic system can be regarded as n×n (n≥2) hyperbolic system with relaxation. We first prove the existence of traveling wave by Gershgorin circle theorem and mathematically describe the neuronal signal transport. Then for a special case n=2, we show the traveling wave is nonlinearly stable, and obtain the convergence rate simultaneously by a weighted estimate.