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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.

The paper deals with the controllability of a degenerate beam equation. In particular, we assume that the left end of the beam is fixed, while a suitable control f$$ f $$ acts on the right end of it. As a first step, we prove the existence of a solution for the homogeneous problem, then we prove some estimates on its energy. Thanks to them, we prove an observability inequality, and using the notion of solution by transposition, we prove that the initial problem is null controllable. [ABSTRACT FROM AUTHOR]

We consider a beam equation in the presence of a leading degenerate operator which is not in divergence form. We impose clamped conditions where the degeneracy occurs and dissipative conditions at the other endpoint. We provide some conditions for the uniform exponential decay of solutions for the associated problem. [ABSTRACT FROM AUTHOR]

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

This article concerns the null controllability of a coupled system of two degenerate parabolic integro-differential equations with one locally distributed control force. Since the memory terms do not allow applying the standards Carleman estimates directly, we start by proving a null controllability result for an associated nonhomogeneous degenerate coupled system employing new Carleman estimates with appropriate weight functions. As a consequence, we deduce the null controllability result for the initial memory system by using the Kakutani’s fixed point Theorem. [ABSTRACT FROM AUTHOR]

In this paper we deal with the null controllability for degenerate/singular parabolic systems with memory terms. To this aim, we first prove the null controllability property for some auxiliary nonhomogeneous degenerate/singular problems via new Carleman estimates for their corresponding adjoint systems. Then, under a condition on the kernels, using the Kakutani's fixed point theorem, we deduce null controllability results for the initial problems with memory. [ABSTRACT FROM AUTHOR]

We consider a differential operator of order 2n of the type Anu = (−1)n(au(n))(n), where a(x) > 0 in [0,1]∖{x0}and a(x0) = 0. We show that, for any n ∈ N, the operator −An generates a contractive analytic semigroup of angle π/2 on L2(0,1). Note that the domain of An depends on the type of degeneracy of a. Our theorems extend some previous results in [3] where n = 1. [ABSTRACT FROM AUTHOR]

We consider a system for a generalized Schnakenberg model, showing Turing pattern formation for a wide class of nonlinearities. [ABSTRACT FROM AUTHOR]

We establish Hardy - Poincaré and Carleman estimates for non-smooth degenerate/singular parabolic operators in divergence form with Neumann boundary conditions. The degeneracy and the singularity occur both in the interior of the spatial domain. We apply these inequalities to deduce well-posedness and null controllability for the associated evolution problem. [ABSTRACT FROM AUTHOR]

We consider a nonlinear elliptic equation with Robin boundary condition driven by the p-Laplacian and with a reaction term which depends also on the gradient. By using a topological approach based on the Leray- Schauder alternative principle, we show the existence of a smooth solution. [ABSTRACT FROM AUTHOR]

We prove a null controllability result for a parabolic problem with Neumann boundary conditions. We consider non smooth coefficients in presence of a strongly singular potential and a strongly degenerate coefficient, both vanishing at an interior point. This paper concludes the study of the Neumann case. [ABSTRACT FROM AUTHOR]