Convergence of Time-Dependent Turing Structures

TL;DR

This paper introduces a novel approach using differential inequalities for analyzing the stability of time-dependent Turing structures, especially useful when traditional linearization methods are inapplicable.

Contribution

It presents a new method based on differential inequalities for stability analysis of parabolic equations with time-dependent coefficients, expanding beyond classical linear stability techniques.

Findings

New stability criteria for time-dependent Turing structures

Applicable to equations with non-constant coefficients

Provides alternative analysis when linearization fails

Abstract

Stability of stationary solutions of parabolic equations is conventionally studied by linear stability analysis, Lyapunov functions or lower and upper functions. We discuss here another approach based on differential inequalities written for the $L^2$ norm of the solution. This method is appropriate for the equations with time dependent coefficients. It yields new results and is applicable when the usual linearization method is not applicable.