REACTION-DIFFUSION PROBLEMS ON TIME-DEPENDENT RIEMANNIAN MANIFOLDS: STABILITY OF PERIODIC SOLUTIONS

We investigate the stability of time-periodic solutions of semilinear parabolic problems with Neumann boundary conditions, posed on a domain of a Riemannian manifold. On the domain we consider metrics that vary periodically in time. The discussion is based on the principal eigenvalue of periodic parabolic operators. The study is related to biological models on the effect of growth and curvature on pattern formation. Metric properties, for instance, the Ricci curvature, play a crucial role.