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Zbl 0779.35049
Dancer, E.N.
On the existence of two-dimensional invariant tori for scalar parabolic equations with time periodic coefficients.
(English)
[J] Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 18, No.3, 455-471 (1991). ISSN 0391-173X

The author considers the equation $$\partial u/\partial t=\Delta u+ g(t,x,u) \text{ in }\Omega\times [0,\infty), \qquad u=0 \text{ on } \partial\Omega \times [0,\infty),$$ where $g:[0,\infty)\times \overline{\Omega}\times \bbfR\to \bbfR$ is $T$-periodic in $t$. If $g$ is independent of $t$, then there is a Lyapunov functional for the semiflow and thus, under very general hypotheses, any solution bounded in a suitable norm for $t\geq 0$ converges to the set of stationary solutions as $t\to\infty$. If $g$ depends $T$-periodically on time, the natural analog of the stationary solutions are the solutions which are $T$- periodic in $t$. Thus the natural conjecture is that, if $g$ is $T$- periodic in $t$, then every solution, which is bounded for $t\geq 0$, must approach that $T$-periodic solutions as $t\to\infty$. The main result of this paper is that this conjecture is false if $\dim \Omega>1$ or if $\Omega$ is 1-dimensional and we use periodic (in $x$) boundary conditions.
[V.Lakshmikantham (Melbourne / Florida)]
MSC 2000:
*35K40 Systems of parabolic equations, general
35B10 Periodic solutions of PDE

Keywords: scalar parabolic equations with time periodic coefficients; periodic solution

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