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).
Similar content being viewed by others
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to Giuseppe Da Prato on the occasion of his 70th birthday
Rights and permissions
About this article
Cite this article
Martinez, P., Vancostenoble, J. Carleman estimates for one-dimensional degenerate heat equations. J. evol. equ. 6, 325–362 (2006). https://doi.org/10.1007/s00028-006-0214-6
Issue Date:
DOI: https://doi.org/10.1007/s00028-006-0214-6