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| This talk is concerned with a unified approach to fast-reaction limits of systems of either two reaction-diffusion equations, or one reaction-diffusion equation and one ODE, on unbounded domains. The equations considered have the form
$$
(P^k) \left\{\begin{array}{rlr}\displaystyle
u_t&= d_u u_{xx} -kF(u,v),
\\
\displaystyle v_t&= d_v v_{xx} -kF(u,v),\\
u(x,0)&=u^k_0(x),\quad v(x,0)=v^k_0(x),
\end{array}\right.$$
where $x \in D = \mathbb{R}$ or $[0, \infty)$, $t \in (0, T)$, the initial data $u^k_0(x)$ and $v^k_0(x)$ are non-negative and bounded, and in the case when $D=[0, \infty)$, $u(0,t) = U_0$ and $v_x(0,t) =0$. Such systems can arise, for example, in modelling fast chemical reactions where there could be either two mobile reactants ($d_u>0$ and $d_v>0$) or a mixture of mobile and immobile reactants ($d_u>0$ and $d_v=0$). The function $F$ is non-negative and increasing in both components, the positive parameter $k$ controls the rate of interaction, and interest is in non-negative solutions because $u$ and $v$ typically correspond to concentrations. Under appropriate conditions on the initial data $u^k_0$ and $v^k_0$, we show that solutions $(u^k, v^k)$ of $(P^k)$ converge in the fast-reaction limit as $ k \to \infty$ to a self-similar limit that has one of four forms, depending on whether $d_v>0$ or $d_v=0$ and on whether $D= \mathbb{R}$ or $D=[0, \infty)$. |
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