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In this paper we study the controllability and the stability for a degenerate beam equation in divergence form via the energy method. The equation is clamped at the left end and controlled by applying a shearing force or a damping at the right end.

In this paper we consider the fourth order operators A1u := (au")" and A2u := au"" in divergence form and non divergence form, respectively, where a, defined in [0, 1] with values in R+, degenerates in an interior point of the interval. Using the semigroup technique, under suitable assumptions on a, we study the generation property of these operators associated to generalized Wentzell boundary conditions, proving the well posedness of the corresponding parabolic problems.

In this paper we focus on the null controllability problem for the heat equation with the so-called inverse square potential and a memory term. To this aim, we first establish the null controllability for a nonhomogeneous singular heat equation by a new Carleman inequality with weights which do not blow up at t=0. Then the null controllability property is proved for the singular heat equation with memory under a condition on the kernel, by means of Kakutani's fixed-point Theorem., Comment: 22 pages

In this paper we study well-posedness and asymptotic stability for a class of nonlinear second-order evolution equations with intermittent delay damping. More precisely, a delay feedback and an undelayed one act alternately in time. We show that, under suitable conditions on the feedback operators, asymptotic stability results are available. Concrete examples included in our setting are illustrated. We give also stability results for an abstract model with alternate positive-negative damping, without delay.

We study the existence of non-trivial, non-negative periodic solutions for systems of singular-degenerate parabolic equations with nonlocal terms and satisfying Dirichlet boundary conditions. The method employed in this paper is based on the Leray-Schauder topological degree theory. However, verifying the conditions under which such a theory applies is more involved due to the presence of the singularity. The system can be regarded as a possible model of the interactions of two biological species sharing the same isolated territory, and our results give conditions that ensure the coexistence of the two species., Comment: 39 pages

We prove the equivalence of the well-posedness of a partial differential equation with nonautonomous past and an associated abstract Cauchy problem. This equivalence is then used to derive sufficient conditions for exponential stability of the solutions.