Generation and propagation of interfaces in reaction-diffusion systems

Abstract

This paper is concerned with the asymptotic behavior, as ε ↘ 0 \varepsilon \searrow 0 , of the solution ( u ε , v ε ) ({u^\varepsilon },{v^\varepsilon }) of the second initial-boundary value problem of the reaction-diffusion system: \[ { u t ε − ε Δ u ε = 1 ε f ( u ε , υ ε ) ≡ 1 ε [ u ε ( 1 − u ε 2 ) − υ ε ] , υ t ε − Δ υ ε = u ε − γ υ ε \left \{ {\begin {array}{*{20}{c}} {u_t^\varepsilon - \varepsilon \Delta {u^\varepsilon } = \frac {1}{\varepsilon }f({u^\varepsilon },{\upsilon ^\varepsilon }) \equiv \frac {1}{\varepsilon }[{u^\varepsilon }(1 - {u^{\varepsilon 2}}) - {\upsilon ^\varepsilon }],} \hfill \\ {\upsilon _t^\varepsilon - \Delta {\upsilon ^\varepsilon } = {u^\varepsilon } - \gamma {\upsilon ^\varepsilon }} \hfill \\ \end {array} } \right . \] where γ > 0 \gamma > 0 is a constant. When v ∈ ( − 2 3 / 9 , 2 3 / 9 ) v \in ( - 2\sqrt 3 /9,2\sqrt 3 /9) , f f is bistable in the sense that the ordinary differential equation u t = f ( u , v ) {u_t} = f(u,v) has two stable solutions u = h − ( v ) u = {h_ - }(v) and u = h + ( v ) u = {h_ + }(v) and one unstable solution u = h 0 ( v ) u = {h_0}(v) , where h − ( v ) , h 0 ( v ) {h_ - }(v), {h_0}(v) , and h + ( v ) {h_ + }(v) are the three solutions of the algebraic equation f ( u , v ) = 0 f(u,v) = 0 . We show that, when the initial data of v v is in the interval ( − 2 3 / 9 , 2 3 / 9 ) ( - 2\sqrt 3 /9,2\sqrt 3 /9) , the solution ( u ε , v ε ) ({u^\varepsilon },{v^\varepsilon }) of the system tends to a limit ( u , v ) (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 u is a "phase" function in the sense that it coincides with h + ( v ) {h_ + }(v) in one region Ω + {\Omega _ + } and with h − ( v ) {h_ - }(v) in another region Ω − {\Omega _ - } . The common boundary (free boundary or interface) of the two regions Ω − {\Omega _ - } and Ω + {\Omega _ + } moves with a normal velocity equal to V ( v ) \mathcal {V}(v) , where V ( ∙ ) \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 ( ∙ , 0 ) − h 0 ( v ( ∙ , 0 ) ) u( \bullet , 0) - {h_0}(v( \bullet , 0)) takes both positive and negative values, then an interface will develop in a short time O ( ε | ln ⁡ ε | ) O(\varepsilon |\ln \varepsilon |) near the hypersurface where u ( x , 0 ) − h 0 ( v ( x , 0 ) ) = 0 u(x,0) - {h_0}(v(x,0)) = 0 .