A permutation characterization of Sturm global attractors of Hamiltonian type
This paper provides a complete characterization of the global attractors of the equation \(u_t=u_{xx}+f(u)\) defined on an interval with Neumann boundary conditions, under a growth condition on \(f\) ensuring that it defines a dissipative dynamical system, and the assumption, which holds generically, that the equilibria are hyperbolic. In a previous work by the first two authors [Trans. Am. Math. Soc. 352, No. 1, 257--284 (2000; Zbl 0932.37065)], more general equations with \(f\) depending also on \(x\) and \(u_x\) were considered, and it was proved that the global attractor is determined by the Sturm permutation corresponding to \(f\), a permutation in \(S_n\), where \(n\) is the number of equilibria, which is defined solely in terms of the equilibria, that is, solutions of the ODE \(u_{xx}+f(x,u,u_x)=0\) satisfying the boundary conditions. This paper determines which permutations are Sturm permutations corresponding to nonlinearities \(f\) which depend only on \(u\). A purely combinatorial characterization of these permutations is proved.
scientific article; zbMATH DE number 5986525
scientific article