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Journal of Evolution Equations

, Volume 6, Issue 2, pp 161–204 | Cite as

Carleman estimates for degenerate parabolic operators with applications to null controllability

  • F. Alabau-BoussouiraEmail author
  • P. Cannarsa
  • G. Fragnelli
Original Paper

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.

Mathematics Subject Classification (2000).

35K65 93B05 93B07 

Keywords.

Degenerate parabolic equations mill controllability observability Carleman estimates Hardy inequality 

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Copyright information

© Birkhäuser Verlag, Basel 2006

Authors and Affiliations

  • F. Alabau-Boussouira
    • 1
    Email author
  • P. Cannarsa
    • 2
  • G. Fragnelli
    • 3
  1. 1.L.M.A.M., CNRS-UMR 7122Université de MetzMetz cedex 01France
  2. 2.Dipartimento di MatematicaUniversitá di Roma “Tor Vergata”RomaItaly
  3. 3.Dipartimento di Ingegneria dell’InformazioneUniversitá di SienaSienaItaly

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