Elliptic and parabolic singular perturbations in the kinetic theory of enzymes (Q1098991)

This paper is devoted to the study of limit-solutions for \(\epsilon\to 0\) of the semilinear parabolic equation \[ \partial u_{\epsilon}/\partial t+A(x,t,\partial /\partial x)u_{\epsilon}+\lambda f(u_{\epsilon}/\epsilon)=g(x,t)\quad (x,t)\in Q, \] \[ u(x,0)=\psi (x),\quad x\in \bar U;\quad \pi_ 0u(x',t)={\hat \psi}(x',t),\quad (x',t)\in \Gamma, \] where \(\lambda\) and \(\epsilon\) are positive parameter and f is a piecewise continuously differentiable function. Such problems occur, for example, in kinetic theory of membranes with enzymotic activity: the parameter \(\lambda\) denotes the ratio of the initial concentrations of the enzymes and the substratum. We denote ``reduced problem'' the problem defined by \(\epsilon =0\). The authors prove that there exists a limit value of \(\lambda\), say \(\lambda_ c\) such that: (i) if \(\lambda <\lambda_ c\) the original problem is a regular perturbation of the original one, and (ii) if \(\lambda >\lambda_ c\), the problem i: [For the entire collection see Zbl 0597.00017.] The authors discuss mixed problems for a certain class of equations which include, in particular, the classical Sobolev equation [see \textit{S. L. Sobolev}, Izv. Akad. Nauk SSSR, Ser. Mat. 18, 3-50 (1954; Zbl 0055.084)] and Rossby equation.