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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.

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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