Sturm global attractors for $S^1$-equivariant parabolic equations

Bernold Fiedler; Carlos Rocha; Matthias Wolfrum; Bernold Fiedler; Carlos Rocha; Matthias Wolfrum

doi:10.3934/nhm.2012.7.617

Networks and Heterogeneous Media

2012, Volume 7, Issue 4: 617-659. doi: 10.3934/nhm.2012.7.617

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Sturm global attractors for $S^1$-equivariant parabolic equations

1.
Freie Universität Berlin, Institut für Mathematik I, Arnimallee 2-6, D-14195 Berlin
2.
Centro de Análise Matemática, Geometria e Sistemas Dinâmicos, Instituto Superior Técnico, Departamento de Matemática, Universidade Técnica de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa

Received: 01 January 2012
Primary: 34K17; Secondary: 34D45, 34K18.

We consider a semilinear parabolic equation of the form $u_t = u_{xx} + f(u,u_x)$ defined on the circle $x ∈ S^1=\mathbb{R}/2\pi\mathbb{Z}$. For a dissipative nonlinearity $f$ this equation generates a dissipative semiflow in the appropriate function space, and the corresponding global attractor $A_f$ is called a Sturm attractor. If $f=f(u,p)$ is even in $p$, then the semiflow possesses an embedded flow satisfying Neumann boundary conditions on the half-interval $(0,\pi)$. This is due to $O(2)$ equivariance of the semiflow and, more specifically, due to reflection at the axis through $x=0,\pi\in S^1$. For general $f=f(u,p)$, where only $SO(2)$ equivariance prevails, we will nevertheless use the Sturm permutation $\sigma$ introduced for the characterization of Neumann flows to obtain a purely combinatorial characterization of the Sturm attractors $A_f$ on the circle. With this Sturm permutation $\sigma$ we then enumerate and describe the heteroclinic connections of all Morse-Smale attractors $A_f$ with $m$ stationary solutions and $q$ periodic orbits, up to $n:=m+2q \le 9$.
- Global attractors,
- semilinear parabolic equations,
- periodic boundary conditions
Citation: Bernold Fiedler, Carlos Rocha, Matthias Wolfrum. Sturm global attractors for $S^1$-equivariant parabolic equations[J]. Networks and Heterogeneous Media, 2012, 7(4): 617-659. doi: 10.3934/nhm.2012.7.617

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Abstract

We consider a semilinear parabolic equation of the form $u_t = u_{xx} + f(u,u_x)$ defined on the circle $x ∈ S^1=\mathbb{R}/2\pi\mathbb{Z}$. For a dissipative nonlinearity $f$ this equation generates a dissipative semiflow in the appropriate function space, and the corresponding global attractor $A_f$ is called a Sturm attractor. If $f=f(u,p)$ is even in $p$, then the semiflow possesses an embedded flow satisfying Neumann boundary conditions on the half-interval $(0,\pi)$. This is due to $O(2)$ equivariance of the semiflow and, more specifically, due to reflection at the axis through $x=0,\pi\in S^1$. For general $f=f(u,p)$, where only $SO(2)$ equivariance prevails, we will nevertheless use the Sturm permutation $\sigma$ introduced for the characterization of Neumann flows to obtain a purely combinatorial characterization of the Sturm attractors $A_f$ on the circle. With this Sturm permutation $\sigma$ we then enumerate and describe the heteroclinic connections of all Morse-Smale attractors $A_f$ with $m$ stationary solutions and $q$ periodic orbits, up to $n:=m+2q \le 9$.

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