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The fast reaction limit for a reaction-diffusion system. (English) Zbl 0865.35063
From the authors’ abstract: This article studies the penetration of radio-labeled antibodies into tumorous tissue described by the reaction-diffusion problem: $u_t=u_{xx}-kuv,\quad v_t=-kuv,\quad 0<x<\infty,\quad 0<t<T,$ $u(x,0)=0,\quad v(x,0)=v_0,\quad x>0,\quad u(0,t)=\phi(t),\quad 0<t<T,$ where $$v_0$$ is the initial constant concentration, and $$\phi$$ is the prescribed concentration assumed to be Hölder continuous, positive, and nondecreasing for $$t\geq 0$$. It is shown that as $$k$$ becomes large, the concentration profiles $$u$$ and $$v$$ converge to limit profiles, and a free boundary develops in the concentration profiles. This limiting behavior is closely related to the limiting behavior of solutions for large time. The results are then generalized to a more general reaction-diffusion system.

##### MSC:
 35K57 Reaction-diffusion equations 35B40 Asymptotic behavior of solutions to PDEs 35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
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