[Paper Review] Spreading in space-time periodic media governed by a monostable equation with free boundaries, Part 2: Spreading speed

Weiwei Ding, Yihong Du+1 more|arXiv (Cornell University)|Nov 7, 2016

Mathematical and Theoretical Epidemiology and Ecology Models7 references3 citations

TL;DR

This paper establishes the existence of asymptotic spreading speed for a monostable reaction-diffusion equation with free boundaries in space-time periodic media, using a novel approach that bypasses the need to construct semi-wave solutions. The authors prove that spreading speed is non-decreasing in the boundary condition parameter μ and converges to a limiting speed as μ → ∞, providing a robust framework for analyzing invasion dynamics in heterogeneous environments.

ABSTRACT

This is Part 2 of our work aimed at classifying the long-time behavior of the solution to a free boundary problem with monostable reaction term in space-time periodic media. In Part 1 (see \cite{ddl}) we have established a theory on the existence and uniqueness of solutions to this free boundary problem with continuous initial functions, as well as a spreading-vanishing dichotomy. We are now able to develop the methods of Weinberger \cite{w1, w2} and others \cite{fyz,lyz,lz1,lz2,lui} to prove the existence of asymptotic spreading speed when spreading happens, without knowing a priori the existence of the corresponding semi-wave solutions of the free boundary problem. This is a completely different approach from earlier works on the free boundary model, where the spreading speed is determined by firstly showing the existence of a corresponding semi-wave. Such a semi-wave appears difficult to obtain by the earlier approaches in the case of space-time periodic media considered in our work here.

Motivation & Objective

• To determine the asymptotic spreading speed in space-time periodic media governed by a monostable reaction-diffusion equation with free boundaries.
• To develop a method that avoids the prior requirement of constructing semi-wave solutions, which is difficult in periodic media.
• To prove the existence and monotonicity of spreading speed with respect to the boundary condition parameter μ.
• To establish the convergence of spreading speed to a limiting value as μ → ∞.
• To extend Weinberger’s method for Cauchy problems to free boundary problems in periodic environments.

Proposed method

• Adapts Weinberger’s approach from [16] to free boundary problems, using order-preserving operators and iterative approximation schemes.
• Employs a sequence of approximating problems with truncated domains and constructs a family of functions $ a^{c}_{n, u}( heta,x) $ to track front propagation.
• Uses the comparison principle and convergence results in Hölder spaces to establish uniform convergence of solutions as μ → ∞.
• Applies the limit operator $ \bar{U} $ to derive the limiting spreading speed $ \bar{c}_+^* $, which serves as the upper bound for $ c^*_{+, u} $.
• Proves monotonicity of $ c^*_{+, u} $ in ν by comparing solutions under different ν values using the order-preserving property.
• Establishes convergence of $ c^*_{+, u} $ to $ \bar{c}_+^* $ via contradiction, showing that any smaller limit would violate the comparison principle.

Experimental results

Research questions

• RQ1 Can spreading speed be established in space-time periodic media without prior knowledge of semi-wave solutions?
• RQ2 What is the behavior of the spreading speed as the boundary condition parameter μ increases?
• RQ3 Does the spreading speed converge to a limiting value as μ → ∞?
• RQ4 How does the spreading speed depend on the parameter μ in the Stefan-type boundary conditions?
• RQ5 Can the method of order-preserving operators be extended to free boundary problems with periodic coefficients?

Key findings

• The asymptotic spreading speed exists for the free boundary problem in space-time periodic media, even without prior knowledge of semi-wave solutions.
• The spreading speed $ c^*_{+, u} $ is non-decreasing in the parameter ν > 0, indicating that stronger boundary feedback leads to faster spreading.
• The limit $ \lim_{\nu \to \infty} c^*_{+, u} = \bar{c}^*_+ $ exists and corresponds to the spreading speed of a related problem with a different boundary condition.
• The rightward spreading speed $ c^*_{+, u} $ converges to $ \bar{c}^*_+ $, which is the spreading speed of the limiting problem (1.7), confirming the robustness of the method.
• The leftward spreading speed $ c^*_{-, u} $ is also non-decreasing and converges to $ \bar{c}^*_- $, the corresponding speed in the limiting problem.
• The proof establishes that $ c^*_{+, u} \leq \bar{c}^*_+ $ for all ν > 0, and equality in the limit as ν → ∞ is confirmed via contradiction.