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Generation and propagation of interfaces in reaction-diffusion systems


Author: Xinfu Chen
Journal: Trans. Amer. Math. Soc. 334 (1992), 877-913
MSC: Primary 35R35; Secondary 35K57
DOI: https://doi.org/10.1090/S0002-9947-1992-1144013-3
MathSciNet review: 1144013
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Abstract: This paper is concerned with the asymptotic behavior, as $ \varepsilon \searrow 0$, of the solution $ ({u^\varepsilon },{v^\varepsilon })$ of the second initial-boundary value problem of the reaction-diffusion system:

$\displaystyle \left\{ {\begin{array}{*{20}{c}} {u_t^\varepsilon - \varepsilon \... ...varepsilon } - \gamma {\upsilon ^\varepsilon }} \hfill \\ \end{array} } \right.$

where $ \gamma > 0$ is a constant. When $ v \in ( - 2\sqrt 3 /9,2\sqrt 3 /9)$, $ f$ is bistable in the sense that the ordinary differential equation $ {u_t} = f(u,v)$ has two stable solutions $ u = {h_ - }(v)$ and $ u = {h_ + }(v)$ and one unstable solution $ u = {h_0}(v)$, where $ {h_ - }(v), {h_0}(v)$, and $ {h_ + }(v)$ are the three solutions of the algebraic equation $ f(u,v) = 0$. We show that, when the initial data of $ v$ is in the interval $ ( - 2\sqrt 3 /9,2\sqrt 3 /9)$, the solution $ ({u^\varepsilon },{v^\varepsilon })$ of the system tends to a limit $ (u,v)$ which is a solution of a free boundary problem, as long as the free boundary problem has a unique classical solution. The function $ u$ is a "phase" function in the sense that it coincides with $ {h_ + }(v)$ in one region $ {\Omega _ + }$ and with $ {h_ - }(v)$ in another region $ {\Omega _ - }$. The common boundary (free boundary or interface) of the two regions $ {\Omega _ - }$ and $ {\Omega _ + }$ moves with a normal velocity equal to $ \mathcal{V}(v)$, where $ \mathcal{V}( \bullet )$ is a function that can be calculated. The local (in time) existence of a unique classical solution to the free boundary problem is also established. Further we show that if initially $ u( \bullet, 0) - {h_0}(v( \bullet, 0))$ takes both positive and negative values, then an interface will develop in a short time $ O(\varepsilon \vert\ln \varepsilon \vert)$ near the hypersurface where $ u(x,0) - {h_0}(v(x,0)) = 0$.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1992-1144013-3
Keywords: Reaction-diffusion systems, generation of interface, propagation of interface
Article copyright: © Copyright 1992 American Mathematical Society

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