On the convergence rate of the nonlinear-hyperbolic systems for axonal transport

TL;DR

This paper analyzes the convergence rate of nonlinear hyperbolic systems modeling axonal transport, showing solutions approach equilibrium at a rate of O(√δ) as relaxation time diminishes.

Contribution

It establishes a convergence rate for BV-solutions of nonlinear hyperbolic systems towards equilibrium in the context of axonal transport models.

Findings

Solutions converge at rate O(√δ) in L1 norm as δ→0.

The convergence is for entropy-satisfying BV-solutions.

Optimality of the convergence rate remains unconfirmed.

Abstract

In this paper, we consider a class of nonlinear reaction-hyperbolic systems with relaxation terms as models for axonal transport in neuroscience. We show the Kruzkov entropy-satisfying BV-solutions of the systems converge towards the solution of an equilibrium model at the rate of $O(\sqrt{\delta})$ in L1 norm as the relaxation time $\delta$ tends to zero. But we don't make sure the rate is optimal.