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Stability in chemical reactor theory. (English) Zbl 1004.35118

Lumer, Günter (ed.) et al., Evolution equations and their applications in physical and life sciences. Proceeding of the Bad Herrenalb (Karlsruhe) conference, Germany, 1999. New York, NY: Marcel Dekker. Lect. Notes Pure Appl. Math. 215, 421-433 (2001).
From the introduction: Let \(D\) be a bounded domain of Euclidean space \(\mathbb{R}^N\), \(N\geq 2\), with smooth boundary \(\partial D\). We consider the following semilinear elliptic boundary value problem arising in chemical reactor theory: \[ \begin{aligned} Au: =\bigl(-\Delta+ c(x)\bigr)u =\lambda\exp \left[{u \over 1+\varepsilon u}\right] & \text{ in }D,\\ Bu:={\partial u\over\partial{\mathbf n}}+ b(x')u=0 & \text{ on }\partial D,\tag{1}\end{aligned} \] where \(c(x)\in C^\infty (\overline D)\) and \(c(x)\geq 0\) in \(D\), \(\lambda\) and \(\varepsilon\) are positive parameters, \(b(x')\in C^\infty(\partial D)\) and \(b(x')\geq 0\) on \(\partial D\), \({\mathbf n}=(n_1,n_2, \dots,n_N)\) is the unit exterior normal to the boundary \(\partial D\).
Let \(\varphi(x)\in C^\infty (\overline D)\) be a unique positive solution of the linearized problem \[ A\varphi=1\text{ in }D,\quad B \varphi=0 \text{ on }\partial D.\tag{2} \] Moreover, we consider the following semilinear parabolic initial-boundary value problem: \[ \begin{aligned} {\partial v\over \partial t} +Av=\lambda \exp\left[{v\over 1+\varepsilon v}\right] & \text{ in }D\times (0, \infty),\\ v|_{t=0}=u_0 & \text{ in }D,\tag{3}\\ Bv:={\partial v \over\partial{\mathbf n}}+b(x')v=0 & \text{ on }\partial D\times (0,\infty).\end{aligned} \] The purpose of this paper is to study the asymptotic stability of maximal and minimal positive solutions of problem (1) in terms of the size of initial values \(u_0\) of problem (3) with respect to the solution \(\varphi(x)\) of problem (2).
For the entire collection see [Zbl 0957.00037].

MSC:

35Q80 Applications of PDE in areas other than physics (MSC2000)
35B35 Stability in context of PDEs
80A32 Chemically reacting flows
35K55 Nonlinear parabolic equations
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