Carleman estimates for one-dimensional degenerate heat equations (original) (raw)

2006, Journal of Evolution Equations

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 + (a(x)w x) x = f, (t, x) ∈ (0, T) × (0, 1), where the function a mainly satisfies a ∈ C 0 ([0, 1]) ∩ C 1 ((0, 1)), a > 0 on (0, 1) and 1 √ a ∈ L 1 (0, 1). 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) = x α (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 = hχ ω , (t,x) ∈ (0, T) × (0, 1), ω ⊂⊂ (0, 1). The present paper completes and improves previous works [7, 8] where this problem was solved in the case a(x) = x α with α ∈ [0, 2).