Spectral instability of small-amplitude periodic waves for hyperbolic non-Fickian diffusion advection models with logistic source

TL;DR

This paper investigates the spectral stability of small-amplitude periodic traveling waves in a hyperbolic non-Fickian diffusion model with logistic source, revealing their emergence from Hopf bifurcation and their inherent spectral instability.

Contribution

It introduces the analysis of spectral stability for periodic waves in a hyperbolic diffusion model with logistic source, highlighting their instability and bifurcation origin.

Findings

Periodic waves emerge from Hopf bifurcation.

Small-amplitude waves are spectrally unstable.

Floquet spectrum analysis shows instability in the linearized operator.

Abstract

A hyperbolic model for diffusion, nonlinear transport (or advection) and production of a scalar quantity, is considered. The model is based on a constitutive law of Cattaneo-Maxwell type expressing non-Fickian diffusion by means of a relaxation time relation. The production or source term is assumed to be of logistic type. This paper studies the existence and spectral stability properties of spatially periodic traveling wave solutions to this system. It is shown that a family of subcharacteristic periodic waves emerges from a local Hopf bifurcation around a critical value of the wave speed. These waves have bounded fundamental period and small-amplitude. In addition, it is shown that these waves are spectrally unstable as solutions to the hyperbolic system. For that purpose, it is proved that the Floquet spectrum of the linearized operator around a wave can be approximated by a linear operator whose point spectrum intersects the unstable half plane of complex numbers with positive real part.