Global attractivity for reaction-diffusion systems. The case of nondiagonal diffusion matrices
The authors consider reaction-diffusion systems with cross diffusion of the form \[ \partial_ t{\mathbf u}=D\Delta{\mathbf u}+f(u) \] subject to the boundary condition \(\beta D\partial_ \nu{\mathbf u}=\alpha({\mathbf u}^*-{\mathbf u})\), where \({\mathbf u}^*\) is a given vector in \(\mathbb{R}^ m\), \(\alpha\) and \(\beta\) are nonnegative constants with \(\alpha+\beta>0\) and \(D\) is a positive definite matrix on \(\mathbb{R}^ m\). They study the convergence to a uniform steady state assuming that such a steady state exists for the associated kinetic system. The main tool is LaSalle's invariance principle.
reaction-diffusion systems with cross diffusion
scientific article; zbMATH DE number 480458