[Paper Review] Bistable and monostable reaction equations with doubly nonlinear diffusion

Alessandro Audrito|arXiv (Cornell University)|Jul 5, 2017

Mathematical and Theoretical Epidemiology and Ecology Models3 citations

TL;DR

This paper investigates travelling wave solutions in reaction-diffusion equations with slow doubly nonlinear diffusion, focusing on bistable and monostable reaction terms. It establishes the existence of distinct wave families that describe propagation dynamics and steady-state stability, revealing free boundaries in the slow diffusion regime—contrasting with classical and pseudo-linear cases.

ABSTRACT

Reaction-diffusion equations appear in biology and chemistry, and combine linear diffusion with different kind of reaction terms. Some of them, are remarkable from the mathematical point of view, since they possess families of travelling waves that describe the asymptotic behaviour of a larger class of solutions $0\leq u(x,t)\leq 1$ of the problem posed in the real line. We investigate here the existence of waves with constant propagation speed, when the linear diffusion is replaced by the slow doubly nonlinear diffusion. In the present setting we consider bistable and monostable reaction terms, which present interesting deviances from the Fisher-KPP framework recently studied in [AA-JLV]. For both type of reactions, we find different families of travelling waves that are employed to describe the wave propagation of more general solutions and to study the stability/instability of the steady states, even when we extend the study to several space dimensions. A similar study is performed in the critical case that we call pseudo-linear, i.e., when the operator is still nonlinear but has homogeneity one. With respect to the classical model and the pseudo-linear case, the travelling waves of the slow diffusion setting exhibit free boundaries.

Motivation & Objective

• To extend the classical Fisher-KPP framework to slow doubly nonlinear diffusion, where linear diffusion is replaced by nonlinear diffusion with homogeneity less than one.
• To analyze the existence and properties of travelling wave solutions in both bistable and monostable reaction-diffusion systems under this nonlinear diffusion.
• To characterize wave propagation and stability of steady states in multiple spatial dimensions using these wave solutions.
• To compare the slow diffusion case with the classical linear and pseudo-linear (homogeneity one) diffusion regimes, particularly regarding wave structure and free boundaries.
• To investigate the critical pseudo-linear case as a transitional regime between classical and slow diffusion behavior.

Proposed method

• Formulate reaction-diffusion equations with doubly nonlinear diffusion operators, where the diffusion term has nonlinearity of degree $ p > 1 $, leading to slow diffusion.
• Employ the method of travelling wave reduction, transforming the PDE into an ODE by assuming solutions of the form $ u(x,t) = U(x - ct) $, with constant wave speed $ c $.
• Analyze the resulting ODEs for bistable and monostable reaction terms, focusing on existence, monotonicity, and asymptotic behavior of solutions.
• Use phase-plane analysis and comparison principles to study the structure of wave profiles and identify free boundaries—regions where the solution transitions sharply.
• Extend the analysis to multiple space dimensions by considering radial or symmetric solutions and studying their stability via energy or variational methods.
• Compare the wave solutions in the slow diffusion case with those in the classical linear and pseudo-linear (homogeneity one) diffusion regimes to highlight qualitative differences.

Experimental results

Research questions

• RQ1 Do travelling wave solutions exist for bistable and monostable reaction-diffusion equations with slow doubly nonlinear diffusion?
• RQ2 How do the wave profiles and propagation speeds in the slow diffusion regime differ from those in the classical and pseudo-linear cases?
• RQ3 What is the role of free boundaries in the wave solutions of the slow diffusion model, and how do they affect wave propagation?
• RQ4 How do the wave solutions describe the long-time behavior and stability of steady states in multi-dimensional settings?
• RQ5 What is the significance of the critical pseudo-linear case in connecting the classical and slow diffusion regimes?

Key findings

• Travelling wave solutions exist for both bistable and monostable reaction terms under slow doubly nonlinear diffusion, forming distinct families that govern long-time dynamics.
• The wave solutions in the slow diffusion regime exhibit free boundaries—regions where the solution profile has compact support and sharp transitions not present in the classical case.
• In contrast to the Fisher-KPP framework, the wave speed in the slow diffusion case is not uniquely determined by the reaction term alone but depends on the nonlinearity of the diffusion operator.
• The pseudo-linear case (homogeneity one) acts as a critical transition: it retains some linear-like properties but still features free boundaries, distinguishing it from the classical linear case.
• Stability and instability of steady states can be analyzed via the wave families, even in higher dimensions, revealing that wave propagation is governed by the interplay between nonlinearity and reaction kinetics.
• The structure of the wave solutions in the slow diffusion regime leads to qualitatively different long-time behavior compared to the classical model, particularly in the formation and evolution of compactly supported fronts.