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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Generation and propagation of interfaces in reaction-diffusion systems
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by Xinfu Chen PDF
Trans. Amer. Math. Soc. 334 (1992), 877-913 Request permission


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: \[ \left \{ {\begin {array}{*{20}{c}} {u_t^\varepsilon - \varepsilon \Delta {u^\varepsilon } = \frac {1}{\varepsilon }f({u^\varepsilon },{\upsilon ^\varepsilon }) \equiv \frac {1}{\varepsilon }[{u^\varepsilon }(1 - {u^{\varepsilon 2}}) - {\upsilon ^\varepsilon }],} \hfill \\ {\upsilon _t^\varepsilon - \Delta {\upsilon ^\varepsilon } = {u^\varepsilon } - \gamma {\upsilon ^\varepsilon }} \hfill \\ \end {array} } \right .\] 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 $O(\varepsilon |\ln \varepsilon |)$ near the hypersurface where $u(x,0) - {h_0}(v(x,0)) = 0$.
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Additional Information
  • © Copyright 1992 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 334 (1992), 877-913
  • MSC: Primary 35R35; Secondary 35K57
  • DOI:
  • MathSciNet review: 1144013