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Ornstein-Zernike behavior for the Bernoulli bond percolation on $\mathbb Z^d$ in the supercritical regime
Persistent regional null contrillability for a class of degenerate parabolic equations
| 1. | Dipartimento di Matematica, Università degli Studi di Roma "Tor Vergata", Via della Ricerca Scientifica, 00133 Roma |
| 2. | Laboratoire de Mathématiques MIP, UMR CNRS 5640, Université Paul Sabatier Toulouse III, 118 route de Narbonne, 31 062 Toulouse cedex 4 |
First, we prove that we can drive the solution to rest at time $T$ in a suitable subset of the space domain (regional null controllability). However, unlike for nondegenerate parabolic equations, this property is no more automatically preserved with time. Then, we prove that, given a time interval $(T,T')$, we can control the equation up to $T'$ and remain at rest during all the time interval $(T,T')$ on the same subset of the space domain (persistent regional null controllability). The proofs of these results are obtained via new observability inequalities derived from classical Carleman estimates by an appropriate use of cut-off functions.
With the same method, we also derive results of regional controllability for a Crocco type linearized equation and for the nondegenerate heat equation in unbounded domains.
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