Existence, uniqueness and qualitative properties of the solution of a degenerate nonlinear parabolic system
The author studies nonnegative solutions for systems of quasilinear degenerate parabolic equations of the form \[ \partial_ tu_ i(x,t)=d_ i\Delta \phi_ i(u(x,t))+F_ i(u_ 1,u_ 2)\quad in\quad \Omega \times {\mathbb{R}}^+;\quad u_ i(x,0)=u_{i0}(x)\quad in\quad \Omega \times \{0\}, \] \[ \partial_{\nu}\phi_ i(u(x,t))+\alpha_ i\phi_ i(u(x,t))=0\quad in\quad \partial \Omega \times {\mathbb{R}}^+\quad for\quad i=1,2. \] Here \(\Omega\) is a bounded domain in the n-dimensional Euclidean space with smooth boundary \(\partial \Omega\), \(d_ i\) are positive constants, \(\partial_{\nu}\) denotes the outward normal, \(\alpha_ i\) are smooth, nonnegative functions, and the (smooth) functions \(\phi_ i\), defined for nonnegative arguments, are positive and increasing, with \(\phi '_ i(u)>0\) for \(u>0\), \(\phi '(0)=0.\) Moreover, the (smooth) functions \(F_ i\) satisfy a quasimonotonicity assumption. Using monotonicity methods, the author establishes existence and uniqueness for the solution; comparison results are also proved, which give insight into the large-time asymptotics. These results are exploited to prove the asymptotic convergence to a positive stationary solution for a positive feedback system occurring in applications. The paper develops and applies to systems the techniques employed by Aronson-Crandall- Peletier for a single equation. For related results, obtained in case of pure Dirichlet boundary conditions using similar methods, cf. \textit{R. Dal Passo} and \textit{P. de Mottoni} [Boll. Unione Mat. Ital., VI. Ser., C, Anal. Funz. Appl. 3, 203-231 (1984; Zbl 0552.35031) and Mathematics in Biology and Medicine, Lect. Notes Biomath. 57, 482-487 (1985; Zbl 0572.35035)].
scientific article; zbMATH DE number 4030064
scientific article