具三井位能之反應擴散梯度系統的三相行波解及穩定解之變分研究

Variational Approaches to Three-Phase Standing and Traveling Waves of Reaction-Diffusion-Gradient Systems with a Triple-Well Potential

簡鴻宇(Hung-Yu Chien)

指導教授 ： 陳俊全

國立臺灣大學/理學院/數學研究所/博士(2019年)

https://doi.org/10.6342/NTU201901241

摘要

針對 Allen-Cahn 類型的反應擴散系統，在高維度空間中我們好奇其解的非平 面結構以及多相性。我們目標是在無界區域如全平面或者⻑條狀定義域上找到此 類系統的駐波 (standing wave) 以及行波 (traveling wave) 解。
我們有兩種嘗試，第一種是利用 Γ-收斂理論，將方程式的解寫成一系列變分 問題的解，並討論其收斂性。我們假設了系統的勢 (potential) 滿足一種簡單的對 稱不變性 — 這個條件比過去部分文獻上假設的在特定對稱群下的不變性還要弱， 以及考慮勢能的低點同時也是系統的常數穩定態 (constant equilibrium)，其中相 對稱的兩點之間有唯一的相變穩定解 (stationary phase transition, 連接此兩個相 的一維方程式穩定解)。最後可以得到一個全平面上的穩定解，而前述的唯一條件 保證了最後得到的解不是常數的退化解。
另外考慮同樣的假設，並多假設了此方程之勢能中的單一相點附近是嚴格上 凹，則可以證明存在一「三相行波解」，連結此「單一相點」與前述「兩點間相變 函數的近似解」。最後根據這個解，我們考慮將⻑條狀定義域拉寬成全平面，相對 應到的解也會存在子序列收斂至一全平面解，由對行波速度的估計可以得知速度 會隨著寬度遞減至零，可知最後收斂得到的全平面解亦是一穩定解。

關鍵字

反應擴散方程 ； 行波解 ； 三相行波解 ； 伽馬收斂

並列摘要

In this paper we aim to find standing and traveling wave solutions, i.e. w(z, y) = u(x, y, t) with z = x − ct, to the reaction-diffusion gradient system with a triple- well potential ∂tu = ∆u − ∇W(u) on an entire domain R2 or a cylindrical domain R×(−l,l).
Firstly by the theory of Γ-convergence, standing wave solutions (i.e. stationary solutions) are obtained under a condition that the potential W is invariant under a simple reflection. This symmetry assumption is weaker than the invariance under a general symmetric group, which is assumed by some literatures.
And also, under the same condition on symmetry, via a variational method, we can show the existence of a traveling wave solution that connects the three constant equilibria in an approximate sense on a cylindrical domain. We propose a convexity condition on one of the equilibria of W to ensure the asymptotical convergence to this equilibrium of the traveling wave solutions at z = −∞.

並列關鍵字

reaction diffusion system ； 3-phase traveling wave solution ； Gamma convergence

參考文獻

[1] S. Alama, L. Bronsard, and C. Gui. Stationary layered solutions in R2 for an Allen-Cahn system with multiple well potential. Calculus of Variations and Partial Differential Equations, 5(4):359–390, 1997.

[2] F. Alessio, A. Calamai, and P. Montecchiari. Saddle-type solutions for a class of semilinear elliptic equations. Adv. Differential Equations, 12(4):361– 380, 2007.

[3] N. D. Alikakos. On the structure of phase transition maps for three or more coexisting phases. In Geometric Partial Differential Equations proceedings, pages 1–31, Pisa, 2013. Scuola Normale Superiore.

[4] N. D. Alikakos and G. Fusco. Entire Solutions to Equivariant Elliptic Sys- tems with Variational Structure. Archive for Rational Mechanics and Analysis, 202(15):567–597, 2011.

[5] N. D. Alikakos and N. I. Katzourakis. Heteroclinic travelling waves of gra- dient diffusion systems. Transactions of the American Mathematical Society, 363(03):1365–1365, 2011.