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- W1984152277 abstract "We consider a semilinear parabolic equation of the form $u_t = u_{xx} + f(u,u_x)$ defined on the circle $x ∈ S^1=mathbb{R}/2pimathbb{Z}$. 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 $x=0,piin S^1$. 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 $n:=m+2q le 9$." @default.
- W1984152277 created "2016-06-24" @default.
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- W1984152277 date "2012-01-01" @default.
- W1984152277 modified "2023-10-16" @default.
- W1984152277 title "Sturm global attractors for $S^1$-equivariant parabolic equations" @default.
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- W1984152277 doi "https://doi.org/10.3934/nhm.2012.7.617" @default.
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