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

Abstract

Abstract:We establish Carleman estimates for singular/degenerate parabolic Dirichlet problems with degeneracy and singularity occurring in the interior of the spatial domain. Our results are completely new, since this situation is not covered by previous contributions for degeneracy and singularity on the boundary. In addition, we consider non-smooth coefficients, thus preventing the use of standard calculations in this framework.

Null controllability for a degenerate population model in divergence form via Carleman estimates

“…There are a lot of papers that deal with null controllability for (1.1) when the dispersion coefficient k is a constant or a strictly positive function (see, for example, [3]). If y is independent of a and k degenerates at the boundary or at an interior point of the domain we refer, for example, to [2], [15] and to [17], [18], [19] if µ is singular at the same point of k. To our best knowledge, [1] is the first paper where y depends on t, a and x and the dispersion coefficient k can degenerate. In particular, the authors assume that k degenerates at the boundary (for example k(x) = x α , being x ∈ (0, 1) and α > 0).…”

Blow-up and delay for a parabolic–elliptic Keller–Segel system with a source term

“…Kozono, Sugiyama, and Takada in [12] considered the problem whether there exists a finite-time self-similar solution of the backward type for the case of N ≥ 2, and Sugiyama and Yahagi in [22] investigated the uniqueness and continuity of weak solutions with respect to the initial data for the Keller-Segel system of degenerate type. For more contribution along this line, we can see [3][4][5][6][19][20][21], and the references therein.…”

Identifying Initial Condition in Degenerate Parabolic Equation with Singular Potential

“…In case we have a priori knowledge of exact 0 in each point of analysis grid-points, we apply the Tikhonov approach to solve the minimization problem (8). The data is assumed to be corrupted by measurement errors, which we will refer to as noise.…”

“…Indeed, many problems coming from physics (boundary layer models in [3], models of Kolmogorov type in [4], etc. ), biology (WrightFisher models in [5] and Fleming-Viot models in [6]), and economics (Black-Merton-Scholes equations in [7]) are described by degenerate parabolic equations [8].…”