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

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


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