We prove an estimate of Carleman type for the one dimensional heat equationwhere a(·) is degenerate at 0. Such an estimate is derived for a special pseudo-convex weight function related to the degeneracy rate of a(·). Then, we study the null controllability on [0, 1] of the semilinear degenerate parabolic equationwhere (t, x) ∈ (0, T ) × (0, 1), ω = (α, β) ⊂⊂ [0, 1], and f is locally Lipschitz with respect to u.

IntroductionIn the last recent years an increasing interest has been devoted to degenerate parabolic equations. Indeed, many problems coming from physics (boundary layer models in [13], models of Kolmogorov type in [7], . . . ), biology (Wright-Fisher models in [50] and Fleming-Viot models in [29]), and economics (Black-MertonScholes equations in [23]) are described by degenerate parabolic equations, whose linear prototype iswith the associated desired boundary conditions, whereIn this paper we concentrate on a special topic related to this field of research, i.e. Carleman estimates for the adjoint problem to (1.1). Indeed, they have so many applications that a large number of papers has been devoted to prove some forms of them and possibly some applications. For example, it is well known that they are a fundamental tool to prove observability inequalities, which lead to global null controllability results for (1.1) also in the non degenerate case: for all T > 0 and for all initial data u 0 ∈ L 2 ((0, T ) × (0, 1)) there is a suitable control h ∈ L 2 ((0, T ) × (0, 1)), supported in a subset ω of [0, 1], such that the solution u of (1.1) satisfies u(T, x) = 0 for all x ∈ [0, 1] (see, for instance, The common point of all the previous papers dealing with degenerate equations, is that the function a degenerates at the boundary of the domain. For example, as a, one can consider the double power functionwhere k and α are positive constants. For related systems of degenerate equations we refer to [1], [2] and [14]. However, the papers cited above deal with a function a that degenerates at the boundary of the spatial domain. To our best knowledge, [51] is the first paper treating the existence of a solution for the Cauchy problem associated to a parabolic equation which degenerates in the interior of the spatial domain, while degenerate parabolic problems modelling biological phenomena and related optimal control problems are later studied in [43] and [9]. Recently, in [32] the authors analyze in detail the degenerate operator A in the space L 2 (0, 1), with or without weight,

We give null controllability results for some degenerate parabolic equations in non divergence form on a bounded interval. In particular, the coefficient of the second order term degenerates at the extreme points of the domain. For this reason, we obtain an observability inequality for the adjoint problem. Then we prove Carleman estimates for such a problem. Finally, in a standard way, we deduce null controllability also for semilinear equations.

We consider a parabolic problem with degeneracy in the interior of\ud the spatial domain, and we focus on Carleman estimates for the\ud associated adjoint problem. The novelty of interior degeneracy does\ud not let us adapt previous Carleman estimate to our situation. As an\ud application, observability inequalities are established

In this paper we study controllability properties of semilinear degenerate parabolic equations. Due to degeneracy, classical null controllability results do not hold in general. Thus we investigate results of 'regional null controllability', showing that we can drive the solution to rest at time T on a subset of the space domain, contained in the set where the equation is nondegenerate.

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.

We give null controllability results for some degenerate parabolic equations in non divergence form with a drift term in one space dimension. In particular, the coefficient of the second order term may degenerate at the extreme points of the space domain. For this purpose, we obtain an observability inequality for the adjoint problem using suitable Carleman estimates

We deal with a degenerate model describing the dynamics of a population depending on time, on age and on space. We assume that the degeneracy can occur at the boundary or in the interior of the space domain and we focus on null controllability problem. To this aim, we prove first Carleman estimates for the associated adjoint problem, then, via cut off functions, we prove the existence of a null control function localized in the interior of the space domain.