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Journal of Evolution Equations

, Volume 6, Issue 2, pp 325–362 | Cite as

Carleman estimates for one-dimensional degenerate heat equations

  • P. MartinezEmail author
  • J. Vancostenoble
Original Paper

Abstract.

In this paper, we are interested in controllability properties of parabolic equations degenerating at the boundary of the space domain.

We derive new Carleman estimates for the degenerate parabolic equation $$ w_t + \left( {a\left( x \right)w_x } \right)_x = f,\quad \left( {t,x} \right) \in \left( {0,T} \right) \times \left( {0,1} \right), $$ where the function a mainly satisfies $$ a \in \mathcal{C}^0 \left( {\left[ {0,1} \right]} \right) \cap \mathcal{C}^1 \left( {\left( {0,1} \right)} \right),a \gt 0 \hbox{on }\left( {0,1} \right) \hbox{and }\frac{1} {{\sqrt a }} \in L^1 \left( {0,1} \right). $$ We are mainly interested in the situation of a degenerate equation at the boundary i.e. in the case where a(0)=0 and / or a(1)=0. A typical example is a(x)=xα (1 − x)β with α, β ∈ [0, 2).

As a consequence, we deduce null controllability results for the degenerate one dimensional heat equation $$ u_t - (a(x)u_x )_x = h\chi _w ,\quad (t,x) \in (0,T) \times (0,1),\quad \omega \subset \subset (0,1). $$

The present paper completes and improves previous works [7, 8] where this problem was solved in the case a(x)=xα with α ∈[0, 2).

Mathematics Subject Classification (2000).

93B05 93C20 93B07 35K65 

Keywords.

Degenerate parabolic equation null controllability Carleman estimates Hardy type inequality 

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Copyright information

© Birkhäuser Verlag, Basel 2006

Authors and Affiliations

  1. 1.Laboratoire de Mathématiques MIP UMR CNRS 5640Université Paul Sabatier Toulouse IIIToulouse Cedex 4France

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