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

Pages: 566 - 575, Issue Special, August 2005

 Abstract        Full Text (231.7K)              

Monica Lazzo - Dipartimento di Matematica, Università di Bari, via Orabona 4, 70125 Bari, Italy (email)
Paul G. Schmidt - Department of Mathematics and Statistics, Parker Hall, Auburn University, AL 36849-5310, United States (email)

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.

Keywords:  Monotone semiflows, invariant manifolds, asymptotic behavior, bistability, semilinear parabolic equations, superlinear growth, subcritical growth.
Mathematics Subject Classification:  Primary: 37C65. Secondary: 35B40, 35K15, 35K20.

Received: September 2004;      Revised: March 2005;      Published: September 2005.