Quasilinear reaction-diffusion systems with nonlinear boundary conditions
This paper analyses the global solvability of the problem \[ \begin{aligned} u_t = \nabla (u^m\nabla u), \;\;v_t = \nabla (v^n\nabla v) \quad &\text{ in } \Omega\times (0,\infty), \\ \frac {\partial u} { \partial \eta} = u^\alpha v^p,\;\;\frac {\partial v}{\partial \eta} = u^qv^\beta \quad &\text{ on }\partial\Omega\times(0,\infty), \\ u(\cdot ,0)=u_0,\;\;v(\cdot,0)=v_0 \quad &\text{ in } \Omega, \end{aligned} \] where \(\Omega\) is a bounded domain in \(\mathbb R^N\) with sufficiently smooth boundary and unit outward normal \(\eta\), \(m\), \(n\),\(\alpha\), and \(\beta\) are nonnegative parameters, and \(p\) and \(q\) are positive parameters. In addition, \(u_0\) and \(v_0\) are positive \(C^1\) functions which satisfy the boundary conditions. It is shown that this problem has a solution (for all time) if and only if \[ \max\{m+\alpha,n+\beta\} \leq 1, \quad pq \leq (1-m-\alpha)(1-n-\beta). \tag{*} \] Previous results were proved in case \(m=n=0\). The proof proceeds by showing that (*) implies the existence of a positive supersolution for all time and that a nonnegative subsolution exists, which becomes infinite in finite time, if (*) does not hold.
scientific article; zbMATH DE number 1274768