Generation and propagation of interfaces in reaction-diffusion systems

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: \[ \left \{ {\begin {array}{*{20}{c}} {u_t^\varepsilon - \varepsilon \Delta {u^\varepsilon } = \frac {1}{\varepsilon }f({u^\varepsilon },{\upsilon ^\varepsilon }) \equiv \frac {1}{\varepsilon }[{u^\varepsilon }(1 - {u^{\varepsilon 2}}) - {\upsilon ^\varepsilon }],} \hfill \\ {\upsilon _t^\varepsilon - \Delta {\upsilon ^\varepsilon } = {u^\varepsilon } - \gamma {\upsilon ^\varepsilon }} \hfill \\ \end {array} } \right .\] 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 $u(x,0) - {h_0}(v(x,0)) = 0$.