Journal of Evolution Equations

, Volume 6, Issue 2, pp 325–362

# Carleman estimates for one-dimensional degenerate heat equations

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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