Stability of singularly perturbed solutions to nonlinear diffusion systems arising in population dynamics (Q1321760)

The author is interested in the existence of spatially inhomogeneous equilibrium solutions of the following reaction diffusion system which is coming from a model equation in a problem of biology: \[ \begin{cases} u_ t = \varepsilon^ 2 u_{xx} + f(u,v) & \text{in \((0,\infty)\times I\)}\\ v_ t = D[(1 + \beta u)v]_{xx} + g(u,v) & \text{in \((0,\infty) \times I\)}\\ u_ x = [(1 + \beta u)v]_ x = 0 & \text{on \((0,\infty) \times \partial I\),}\end{cases}\tag{P} \] where \(f\) and \(g\) satisfy some conditions so that (P) is a competition system. The diffusion term is nonlinear and hence somewhat complicated. Such a thing is called the cross diffusion term which arises because of a certain biological effect. The main question is whether the spatially inhomogeneous equilibrium solution is stable or not. In the case \(\beta = 0\) (no cross diffusion), it is known that such a solution does not exist. This fact was proved by Kishimoto and Weinberger. The author finds a different situation from the case \(\beta = 0\), including the existence of spatially inhomogeneous stable solutions in the singularly perturbation situation, that is, \(\varepsilon\) is very small.