Carleman estimates for one-dimensional degenerate heat equations |
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Authors: | P Martinez J Vancostenoble |
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Institution: | (1) Laboratoire de Mathématiques MIP UMR CNRS 5640, Université Paul Sabatier Toulouse III, 118 route de Narbonne, 31 062 Toulouse Cedex 4, France |
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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).
Dedicated to Giuseppe Da Prato on the occasion of his 70th birthday |
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Keywords: | 93B05 93C20 93B07 35K65 |
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