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On existence, uniqueness and attractivity of stationary solutions to some quasilinear parabolic systems. (English) Zbl 0572.35035

Mathematics in biology and medicine, Proc. Int. Conf., Bari/Italy 1983, Lect. Notes Biomath. 57, 482-487 (1985).
[For the entire collection see Zbl 0559.00025.]
The systems considered here are of the following form, solutions being \(u_ 1(x)\), \(u_ 2(x):\) \(\Delta \phi_ i(u_ 1,u_ 2)+f_ i(u_ 1,u_ 2)=0\) in \(\Omega\), \(i=1,2,\) with homogeneous Dirichlet conditions. First, results from the authors’ paper [Boll. Unione Mat. Ital., VI, Ser. C. Anal. Funz. Appl. 3, 203-231 (1984; Zbl 0552.35031)] are reviewed; they give conditions for existence, uniqueness, and global stability with reference to the corresponding parabolic systems. Then various examples are presented, representing models for the spread of infectious diseases.
Reviewer: P.Fife

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
92D25 Population dynamics (general)
35B35 Stability in context of PDEs
35A05 General existence and uniqueness theorems (PDE) (MSC2000)