Design of Sturm global attractors 1: Meanders with three noses, and reversibility

Abstract

We systematically explore a simple class of global attractors, called Sturm due to nodal properties, for the semilinear scalar parabolic partial differential equation (PDE) u(t) = u(xx) + f(x, u, u(x)) on the unit interval 0 < x < 1, under Neumann boundary conditions. This models the interplay of reaction, advection, and diffusion. Our classification is based on the Sturm meanders, which arise from a shooting approach to the ordinary differential equation boundary value problem of equilibrium solutions u(t) = 0. Specifically, we address meanders with only three "noses," each of which is innermost to a nested family of upper or lower meander arcs. The Chafee-Infante paradigm, with cubic nonlinearity f = f(u), features just two noses. Our results on the gradient-like global PDE dynamics include a precise description of the connection graphs. The edges denote PDE heteroclinic orbits v(1) (sic) v(2) between equilibrium vertices v(1), v(2) of adjacent Morse index. The global attractor turns out to be a ball of dimension d, given as the closure of the unstable manifold W-u(O) of the unique equilibrium with maximal Morse index d. Surprisingly, for parabolic PDEs based on irreversible diffusion, the connection graph indicates time reversibility on the (d - 1)-sphere boundary of the global attractor.