Existence and uniqueness of positive stationary solution to a reaction diffusion system from biology
The system \[ - \Delta u = f(u) - v,\;- \Delta v = \delta u - \gamma v,\;x \in \Omega,\;u = v,x \in \partial \Omega \tag{1} \] where \(\Omega \subset \mathbb{R}^n\) is a bounded open domain, \(\partial \Omega \in C^\alpha\), \(\delta, \gamma > 0\), is interpreted as stationary model for relative concentrations of substances known as morphogens. The author proves, under some conditions concerning the eigenvalues of \(-\Delta\), that the system (1) has a nontrivial and nonnegative solution and that this solution is unique.
scientific article; zbMATH DE number 722356