Similarity Solutions for the Flux limited Keller Segel System with Time Varying Chemical Decay Rate

TL;DR

This paper conducts a symmetry analysis of a flux limited Keller Segel system with time-varying chemical decay, classifying symmetries and deriving explicit solutions to understand chemotaxis with dynamic decay rates.

Contribution

It provides a complete group classification for the PDE system with variable decay rates and constructs similarity solutions, advancing analytical methods for chemotaxis models.

Findings

Identified symmetry algebras for different decay functions.

Derived similarity reductions and explicit solutions.

Classified cases with constant, inverse, and exponential decay.

Abstract

We investigate a one dimensional flux limited Keller Segel system (FLKS) in which the chemical decay rate is allowed to vary explicitly in time, a feature motivated by enzymatic regulation and environmental variability in chemotactic signalling. Treating the decay rate as an arbitrary function, we carry out a systematic Lie symmetry analysis of the resulting PDE system and employ equivalence transformations to perform a complete group classification, we identify the kernel symmetry algebra admitted for arbitrary decay functions and determine three distinguished cases that extend the symmetry algebra constant decay rates, inverse time (power law) decay, and exponential decay. For each case, we construct an optimal system of subalgebras and derive the corresponding similarity reductions. Finally, we find some explicit solutions for our FLKS model. Our results provide a rigorous mathematical foundation for understanding which temporal decay patterns admit similarity reductions, thereby enabling analytical progress on flux limited chemotaxis models with realistic time varying degradation mechanisms.