A relation between cross-diffusion and reaction-diffusion (Q428442)

The following problem NEWLINE\[NEWLINE\begin{cases} \frac{{\partial z_i }} {{\partial t}} = \Delta [a_i z_i + \phi _i (z)] + f_i (z)\,\, &\text{in}\,\,Q: = \Omega \times (0,T),\,i = 1,\dots ,n,\\ a_i z_i + \phi _i (z) = 0\,\, &\text{on}\,\,\partial \Omega \times (0,T),\,\,\,i = 1,\dots ,n,\\ z( \cdot ,0) = z_0 \,\, &\text{in}\,\,\;\Omega \end{cases}NEWLINE\]NEWLINE where \(z \in \mathbb R^n\), \(\Omega \subset \mathbb R^N \) is a bounded domain with smooth boundary \(\partial \Omega\), \(T\) and \(a_i (i = 1,\dots n)\) are positive constants, \(\phi = (\phi _1 ,\dots ,\phi _n )\), \(f = (f_1 ,\dots ,f_n ):\mathbb R^n \to \mathbb R^n\) and \(z_0 = (z_{01} ,\dots ,z_{0n} ):\Omega \to \mathbb R^n \) are given functions. The authors extended results of \textit{M. Iida} and \textit{H. Ninomiya} [Recent advances on elliptic and parabolic issues, in: Proceedings of the 2004 Swiss-Japanese seminar, Zürich, Switzerland 2004. Hackensack, NJ: World Scientific. 145--164 (2006; Zbl 1141.35388)] to a more general cross-diffusion problem involving strongly coupled systems. It is shown that a solution of the problem can be approximated by that of a semilinear reaction-diffusion system without any assumptions on the solutions. This indicates that the mechanism of cross-diffusion might be captured by reaction-diffusion interaction.