Numbers of zeros on invariant manifolds in reaction-diffusion equations. (English) Zbl 0594.35056

Brunovský, Pavol; Fiedler, Bernold

Nonlinear Anal., Theory Methods Appl. 10, 179-193 (1986).

The ”orbit connection” problem for dynamical systems consists in establishing whether, given two (hyperbolic) rest points \(v_ 1\), \(v_ 2\), having a nontrivial unstable, respectively stable manifold, there is a solution u(t) (”orbit”) such that \(u(-\infty)=v_ 1\), \(u(+\infty)=v_ 2\). In the specific case of the dynamical system associated with a one- dimensional semilinear parabolic equation, an insight into this problem can be provided by the analysis of the number of sign changes of the solutions. This is indeed the main motivation for the present paper, in which bounds are established for an (appropriately defined) number of zeros of \(u(t)-v_ 1\) and \(u(t)-v_ 2\). The proofs involve two distinct tools: namely monotonicity methods and, on the other hand, a detailed analysis of the structure of the unstable and of the stable manifold of a hyperbolic stationary solution.

Reviewer: P.de Mottoni

MSC:

35K55	Nonlinear parabolic equations
80A20	Heat and mass transfer, heat flow (MSC2010)
35Q99	Partial differential equations of mathematical physics and other areas of application
35B40	Asymptotic behavior of solutions to PDEs

Keywords:

lap number; zero number; invariant manifolds; reaction-diffusion; equations; orbit connection problem; dynamical systems; one-dimensional semilinear parabolic equation; sign changes of the solutions; monotonicity methods; hyperbolic stationary solution

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