Carleman estimates for degenerate parabolic operators with applications to null controllability |
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Authors: | F Alabau-Boussouira P Cannarsa G Fragnelli |
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Institution: | (1) L.M.A.M., CNRS-UMR 7122, Université de Metz, Ile du Saulcy, 57045 Metz cedex 01, France;(2) Dipartimento di Matematica, Universitá di Roma “Tor Vergata”, Via della Ricerca Scientifica, 00133 Roma, Italy;(3) Dipartimento di Ingegneria dell’Informazione, Universitá di Siena, Via Roma 56, 53100 Siena, Italy |
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Abstract: | We prove an estimate of Carleman type for the one dimensional heat equation
$$ u_t - \left( {a\left( x \right)u_x } \right)_x + c\left( {t,x} \right)u = h\left( {t,x} \right),\quad \left( {t,x} \right)
\in \left( {0,T} \right) \times \left( {0,1} \right), $$ where a(·) is degenerate at 0. Such an estimate is derived for a
special pseudo-convex weight function related to the degeneracy rate of a(·). Then, we study the null controllability on 0,
1] of the semilinear degenerate parabolic equation
$$ u_t - \left( {a\left( x \right)u_x } \right)_x + f\left( {t,x,u} \right) = h\left( {t,x} \right)\chi _\omega \left( x \right),
$$ where (t, x) ∈(0, T) × (0, 1), ω=(α, β) ⊂⊂ 0, 1], and f is locally Lipschitz with respect to u.
Dedicated to Giuseppe Da Prato on the occasion of his 70th birthday |
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Keywords: | 35K65 93B05 93B07 |
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