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

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


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
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[1] Atkinson, F.V., Discrete and continuous boundary problems, (1964), Academic Press · Zbl 0117.05806
[2] Brunovský, P.; Fiedler, B., Simplicity of zeros in scalar parabolic equations, (1984), preprint · Zbl 0549.35062
[3] Hale, J.K., Topics in dynamic bifurcation theory, () · Zbl 0483.58018
[4] H{\scAle} J.K. & do N{\scAscimento} A.S., Orbit connections in a parabolic equation, preprint.
[5] Hartman, P., Ordinary differential equations, (1982), Birkhäuser Boston · Zbl 0125.32102
[6] Henry, D., Geometric theory of semilinear parabolic equations, () · Zbl 0456.35001
[7] Matano, H., Nonincrease of the lap number of a solution for a one dimensional semilinear parabolic equation, Pub. fac. sci. univ. Tokyo, 29, 401-441, (1982), Sec. 1A · Zbl 0496.35011
[8] Protter, M.; Weinberger, H., Maximum principles in differential equations, (1967), Prentice Hall Englewood Cliffs, NJ · Zbl 0153.13602
[9] Redheffer, R.M.; Walter, W., The total variation of solutions of parabolic differential equations and a maximum principle in unbounded domains, Math. annln, 209, 57-67, (1974) · Zbl 0267.35053
[10] Smoller, J., Shock waves and reaction diffusion equations, (1983), Springer New York · Zbl 0508.35002
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