Monotone local semiflows with saddle-point dynamics and applications to semilinear diffusion equations

Monica Lazzo; Paul G. Schmidt

doi:10.3934/proc.2005.2005.566

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2005, Volume 2005: 566-575. Doi: 10.3934/proc.2005.2005.566 open access

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Monotone local semiflows with saddle-point dynamics and applications to semilinear diffusion equations

Monica Lazzo^1, and
Paul G. Schmidt^2,

1.
Dipartimento di Matematica, Università di Bari, via Orabona 4, 70125 Bari
2.
Department of Mathematics and Statistics, Parker Hall, Auburn University, AL 36849-5310

Received: August 31, 2004

Revised: February 28, 2005

Published: August 2005

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Abstract

Consider a monotone local semiflow in the positive cone of a strongly ordered Banach space, for which $0$ and $\infty$ are stable attractors, while all nontrivial equilibria are unstable. We prove that under suitable monotonicity, compactness, and smoothness assumptions, the two basins of attraction, $\Bz$ and $\Bi$, are separated by a Lipschitz manifold $\M$ of co-dimension one that forms the common boundary of $\Bz$ and $\Bi$. This abstract result is applied to a class of semilinear reaction-diffusion equations with superlinear, yet subcritical reaction terms.

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Mathematics Subject Classification: Primary: 37C65. Secondary: 35B40, 35K15, 35K20.

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