2023/06/08 by Fiedler, Bernold, Rocha, Carlos · 1 citation
#35K58 #37D15 (Primary) 35B41 #Analysis of PDEs (math.AP) #Dynamical Systems (math.DS) #FOS: Mathematics
paper · doi:10.48550/arxiv.2306.05232
This sequel continues our exploration arxiv:2302.12531 of a deceptively ``simple'' class of global attractors, called Sturm due to nodal properties. They arise for the semilinear scalar parabolic PDE ut = uxx + f(x,u,ux) 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 ODE boundary value problem of equilibrium solutions u=v(x). 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 of 1974, with cubic nonlinearity f=f(u), features just two noses. We present, and fully prove, a precise description of global PDE connection graphs, graded by Morse index, for such gradient-like Morse-Smale systems \eqrefeq:*. The directed edges denote PDE heteroclinic orbits v1 \leadsto v2 between equilibrium vertices v1, v2 of adjacent Morse index. The connection graphs can be described as a lattice-like structure of Chafee-Infante subgraphs. However, this simple description requires us to adjoin a single ``equilibrium'' vertex, formally, at Morse level -1. Surprisingly, for parabolic PDEs based on irreversible diffusion, the connection graphs then also exhibit global time reversibility.