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.

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 consider operators in divergence and in nondivergence form with degeneracy at the interior of the space domain. Characterizing the domain of the operators, we prove that they generate positive analytic semigroupson spaces of L^2 type.Finally, some applications to linear and semilinear parabolic evolution problems and to linear hyperbolic ones are presented.

The aim of the paper is to provide conditions ensuring the existence of non-trivial non-negative periodic solutions to a system of doubly degenerate parabolic equations containing delayed nonlocal terms and satisfying Dirichlet boundary conditions. The employed approach is based on the theory of the Leray-Schauder topological degree theory, thus a crucial purpose of the paper is to obtain a priori bounds in a convenient functional space, here $L^2(Q_T)$, on the solutions of certain homotopies. This is achieved under different assumptions on the sign of the kernels of the nonlocal terms. The considered system is a possible model of the interactions between two biological species sharing the same territory where such interactions are modeled by the kernels of the nonlocal terms. To this regard the obtained results can be viewed as coexistence results of the two biological populations under different intra and inter specific interferences on their natural growth rates.

In this paper we prove a general result giving the maximum and the antimaximum principles in a unitary way for linear operators of the form $L+\lambda I$, provided that 0 is an eigenvalue of L with associated constant eigenfunctions. To this purpose, we introduce a new notion of "quasi"-uniform maximum principle, named k-uniform maximum principle: it holds for lambda belonging to certain neighborhoods of 0 depending on the fixed positive multiplier k > 0 which selects the good class of right-hand-sides. Our approach is based on a $L^infinity$ - $L^p$ estimate for some related problems. As an application, we prove some generalization and new results for elliptic problems and for time periodic parabolic problems under Neumann boundary conditions