Positively invariant regions for strongly coupled reaction-diffusion systems with a balance law
The present paper studies positively invariant regions for strongly coupled parabolic systems of the form \(\partial_t u_i(x,t)= \sum^m_{j =1} a_{ij}Lu_j (x,t)+F_i(x,t,u_1, \dots,u_m)\) in \(\Omega\times [0,T)\) with initial conditions \(u_i(x,0)= u_{i,0}(x)\), and boundary conditions \(\partial_\nu u_i(x, t)= G_i(x,t,u_1, \dots,u_m)\) defined on \(\partial\Omega\) (here \(i=1,\dots, m)\). The results stated are restricted to \(m=2\). The operator \(L\) is of second order, is uniformly elliptic and equal for each component, the \(a_{ij}\) are constant and the domain \(\Omega\subset \mathbb{R}^n\) is bounded with \(\partial_\nu\) denoting the conormal derivative. A number of hypothesis are formulated such that an algebraic transformation reduces the strongly coupled system to a weakly coupled one for which an earlier comparison result of the author [SIAM J. Math. Anal. 11, 1075-1103 (1980; Zbl 0468.35055)] can be applied. One assumption is the existence of a function \(\varphi\) that satisfies \(L\varphi \geq \alpha\) in \(\Omega\) and \(\partial_\nu \varphi\geq\beta\) on \(\partial\Omega\) and is expected to be positive. The \(\alpha\) and \(\beta\) depend on a ``balance law'' for the nonlinear terms.
scientific article; zbMATH DE number 1533171
scientific article