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We consider two degenerate heat equations with a nonlocal space term, studying, in particular, their null controllability property. To this aim, we first consider the associated nonhomogeneous degenerate heat equations: we study their well posedness, the Carleman estimates for the associated adjoint problems and, finally, the null controllability. Then, as a consequence, using the Kakutani's fixed point Theorem, we deduce the null controllability property for the initial nonlocal problems. [ABSTRACT FROM AUTHOR]

In this paper, we study the null controllability for the problems associated to the operators yt−Ay−λb(x)y+∫01K(t,x,τ)y(t,τ)dτ,(t,x)∈(0,T)×(0,1),$$ {y}_t- Ay-\frac{\lambda }{b(x)}y+\int_0^1K\left(t,x,\tau \right)y\left(t,\tau \right)\kern0.1em d\tau, \kern0.30em \left(t,x\right)\in \left(0,T\right)\times \left(0,1\right), $$where Ay:=ayxx$$ Ay:= a{y}_{xx} $$ or Ay:=(ayx)x$$ Ay:= {\left(a{y}_x\right)}_x $$ and the functions a$$ a $$ and b$$ b $$ degenerate at an interior point x0∈(0,1)$$ {x}_0\in \left(0,1\right) $$. To this aim, as a first step, we study the well‐posedness, the Carleman estimates, and the null controllability for the associated nonhomogeneous degenerate and singular heat equations. Then, using Kakutani's fixed point Theorem, we deduce the null controllability property for the initial nonlocal problems. [ABSTRACT FROM AUTHOR]

In this paper we study the null controllability property for a single population model in which the population y depends on time t, space x, age a and size τ . Moreover, the diffusion coefficient k is degenerate at a point of the domain or both extremal points. Our technique is essentially based on Carleman estimates. The τ dependence requires us to modify the weight for the Carleman estimates, and accordingly the proof of the observability inequality. Thanks to this observability inequality we obtain a null controllability result for an intermediate problem and finally for the initial system through suitable cut off functions. [ABSTRACT FROM AUTHOR]

In this paper, we analyze the null controllability property for a degenerate parabolic equation involving memory terms with a locally distributed control. We first derive a null controllability result for a nonhomogeneous degenerate heat equation via new Carleman estimates with weighted time functions that do not blow up at t=0. Then, this result is successfully used with a classical fixed point to obtain null controllability for the initial memory system. [ABSTRACT FROM AUTHOR]

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. [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 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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We consider non-smooth general degenerate/singular parabolic equations in non-divergence form with degeneracy and singularity occurring in the interior of the spatial domain, in presence of Dirichlet or Neumann boundary conditions. In particular, we consider well posedness of the problem and then we prove Carleman estimates for the associated adjoint problem. [ABSTRACT FROM AUTHOR]

We prove null controllability results for the one dimensional degenerate heat equation in non divergence form with a drift term and Neumann boundary conditions. To this aim we prove Carleman estimates for the associated adjoint problem. Some linear extensions are considered. [ABSTRACT FROM AUTHOR]