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Generic Morse-Smale property for the parabolic equation on the circle. (English) Zbl 1213.35046

The following scalar reaction-diffusion equation on \(S^1\) is considered
\[ \begin{cases} u_t(x,t)= u_{xx}(x,t)+f(x,u(x,t),u_x(x,t)), &(x,t)\in S^1\times(0,\infty),\\ u(x,0)=u_0(x), &x\in S^1, \end{cases} \]
where \(f\) belongs to the space \(C^2(S^1\times\mathbb{R}\times\mathbb{R},\mathbb{R})\), and \(u_0\) is a given function in the Sobolev space \(H^s(S^1)\), \(s\in(3/2,2)\), so that \(H^s(S^1)\) is continuously embedded into \(C^{1+\alpha}\) for \(\alpha=s-3/2\).
On the basis of the lap number property, exponential dichotomies, Sard-Smale theorem and analysis of the asymptotic behaviour of solutions of the linearized equations along the connecting orbits the authors complete their previous results [R. Joly and G. Raugel, Trans. Am. Math. Soc. 362, No. 10, 5189–5211 (2010; Zbl 1205.35151)] and show that any connecting orbit between two hyperbolic equilibria with distinct Morse indices or between a hyperbolic equilibrium and hyperbolic periodic orbit is automatically transverse. Generically, with respect to \(f\), there does not exist any connection between equilibria with the same Morse index. The main result of the article is that generically with respect to \(f\) the nonwandering set consists of a finite number of hyperbolic equilibria and periodic orbits.

MSC:

35B10 Periodic solutions to PDEs
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35K57 Reaction-diffusion equations
37D05 Dynamical systems with hyperbolic orbits and sets
37D15 Morse-Smale systems
37L45 Hyperbolicity, Lyapunov functions for infinite-dimensional dissipative dynamical systems
35B40 Asymptotic behavior of solutions to PDEs

Citations:

Zbl 1205.35151
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References:

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