Interior degenerate/singular parabolic equations in nondivergence form: well-posedness and Carleman estimates

Abstract

We consider non smooth general degenerate/singular parabolic equations in non\ud divergence form with degeneracy and singularity occurring in the interior of the spatial\ud domain, in presence of Dirichlet or Neumann boundary conditions. In particular, we\ud consider well posedness of the problem and then we prove Carleman estimates for the\ud associated adjoint problem

Carleman estimates for parabolic equations with interior degeneracy and Neumann boundary conditions

“…Using again the fact that ξ x is supported in [−β 1 , −λ 1 ]∪[λ 1 ,β 1 ]∪[λ 2 ,λ 2 ]∪[2−λ 2 , 2−λ 2 ] and the boundedness ofã ′ (far away from x 0 in the weakly degenerate case, see (4.2), and since a ∈ W 1,∞ (−1, 2) in the strongly degenerate one), it follows, by the Caccioppoli's inequality for the nondegenerate case (see, e.g., [18,Remark 7]).…”

Carleman estimates for singular parabolic equations with interior degeneracy and non-smooth coefficients

“…Clearly, this result generalizes the one obtained in [29] or [30]: in fact, if λ = 0 (that is, if we consider the purely degenerate case), we recover the main contributions therein. See also [25] for the problem in non divergence form for both Dirichlet and Neumann boundary conditions.…”

Carleman estimates and null controllability for a degenerate population model

“…This necessary condition that ensures the well posedness of (1.1) makes it not null controllable (see [31] for the interior degeneracy). For this reason, in this paper as in [13], [14], [26], [27] or [31], we prove null controllability for (1.1) without deducing it by the previous results for the problem in divergence form. Therefore, this paper complements [2].…”