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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]

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]

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]

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 study the existence of nontrivial, nonnegative 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. [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]

Abstract: We consider a system of two degenerate parabolic equations with nonlocal terms and Dirichlet boundary conditions. More precisely, the degeneracy in each equation of the system is of the type -Laplacian where is a function depending on , where Ω is a bounded smooth domain of . The system models the diffusion and the interaction between two different biological species sharing the same territory Ω. The paper provides conditions on the parameters of the problem that guarantee the coexistence of a T-periodic non-negative solution with both non-trivial . [Copyright &y& Elsevier]