# The reactive-telegraph equation and a related kinetic model

## TL;DR

This paper analyzes the long-term behavior of the reactive-telegraph equation and a related kinetic model, revealing dimension-dependent properties and phase transitions in propagation speed.

## Contribution

It introduces a reactive-kinetic model for higher dimensions and characterizes the propagation speed, including explicit calculations in one and two dimensions.

## Key findings

- Reactive-telegraph equation does not preserve positivity in higher dimensions.
- A phase transition between parabolic and hyperbolic behavior occurs only in one dimension.
- Explicit propagation speeds are computed in one and two dimensions.

## Abstract

We study the long-range, long-time behavior of the reactive-telegraph equation and a related reactive-kinetic model. The two problems are equivalent in one spatial dimension. We point out that the reactive-telegraph equation, meant to model a population density, does not preserve positivity in higher dimensions. In view of this, in dimensions larger than one, we consider a reactive-kinetic model and investigate the long-range, long-time limit of the solutions. We provide a general characterization of the speed of propagation and we compute it explicitly in one and two dimensions. We show that a phase transition between parabolic and hyperbolic behavior takes place only in one dimension. Finally, we investigate the hydrodynamic limit of the limiting problem.