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Carleman estimates for degenerate parabolic operators with applications to null controllability. (English) Zbl 1103.35052
The aim of the paper is to provide a full analysis of the null controllability problem for a linear parabolic equation with degenerate coefficient \(a(x)\) of major derivative of the elliptic operator. Two types of degeneracy for \(a(x)\) are considered, weak and strong degeneracy, each type being associated with its own boundary condition at \(x=0\). The main technical part of the paper is the analysis of the adjoint problem in the weak and strong cases as well. For each case, the Carleman estimates are established. In the conclusive part of the work, conditions of null controllability for a semilinear equation with a Lipschitz right-hand side are obtained with the use of the fixed point method.

35K65 Degenerate parabolic equations
93B05 Controllability
93B07 Observability
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