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In this paper we study the null controllability for the problems associated to the operators y_t-Ay - \lambda/b(x) y+\int_0^1 K(t,x,\tau)y(t, \tau) d\tau, (t,x) \in (0,T)\times (0,1) where Ay := ay_{xx} or Ay := (ay_x)_x and the functions a and b degenerate at an interior point x0 Ë .0; 1/. 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 the Kakutani’s fixed point Theorem, we deduce the null controllability property for the initial nonlocal problems.

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 $$\tau $$ . 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 $$\tau $$ 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.

In this paper we consider a degenerate population equation in divergence form depending on time, on age and on space and we prove a related null controllability result via Carleman estimates., Comment: arXiv admin note: substantial text overlap with arXiv:1611.06024 and arXiv:1902.02655

We consider a system for a generalized Schnakenberg model, showing Turing pattern formation for a wide class of nonlinearities.

This paper is a corrigendum of one hypothesis introduced in Mem. Amer. Math. Soc. 242 (2016), no. 1146, and used again in J. Differential Equations 260 (2016), pp. 1314–1371 and Adv. Nonlinear Anal. 6 (2017), pp. 61–84]. We give here the corrected proofs of the concerned results, improving most of them.

In this paper we consider the null controllability for a population model depending on time, on space and on age. Moreover, the diffusion coefficient degenerate at the boundary of the space domain. The novelty of this paper is that for the first time we consider the presence of a memory term, which makes the computations more difficult. However, under a suitable condition on the kernel we deduce a null controllability result for the original problem via new Carleman estimates for the adjoint problem associated to a suitable nonhomogeneous parabolic equation.

We show Carleman estimates for parabolic problems in divergence or non divergence form with degeneracy at the boundary or in the interior of the space domain. By them we obtain observability inequalities, proving that the problems are null controllable.

We consider parabolic problems in divergence and non divergence form with interior degeneracy and singularity given by general functions, showing well posedness and null controllability.

We consider parabolic problems in divergence form with boundary degeneracy and power singularity, showing well posedness and null controllability.

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.

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., 29 pages

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.

We establish Hardy - Poincare 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.

We consider a parametric nonlinear Robin problem driven by the $p -$Laplacian plus an indefinite potential and a Caratheodory reaction which is $(p-1) -$ superlinear without satisfying the Ambrosetti - Rabinowitz condition. We prove a bifurcation-type result describing the dependence of the set of positive solutions on the parameter. We also prove the existence of nodal solutions. Our proofs use tools from critical point theory, Morse theory and suitable truncation techniques.

In this paper we consider a nonlinear elliptic problem driven by a nonhomogeneous differential operator with Robin boundary conditions. We produce conditions on the reaction term near 0 + ${0^{+}}$ and near + ∞ ${+\infty}$ , which imply the existence and uniqueness of a positive solution. In the particular case of equations driven by the p-Laplacian, we show that these conditions are also necessary, extending in this way the semilinear Dirichlet work of Brezis and Oswald [5].

We consider operators in divergence form, $A_1u=(au')'$, and in nondivergence form, $A_2u=au''$, provided that the coefficient $a$ vanishes in an interior point of the space domain. Characterizing the domain of the operators, we prove that, under suitable assumptions, the operators $A_1$ and $A_2$, equipped with general Wentzell boundary conditions, are nonpositive and selfadjoint on spaces of $L^2$ type.

In this paper we consider a system in non divergence form which models the interaction between two different species u and v . Both of them depend on time, on age and on space. Moreover, the diffusion coefficients degenerate at the boundary of domain. We study, in particular, null controllability of the system via observability inequalities and Carleman estimates for the associated non homogeneous adjoint problem.

We consider a parabolic problem with degeneracy in the interior of the spatial domain and we focus on the well-posedness of the problem and on inverse source problems. The novelties of the present paper are two. First, the degeneracy point is in the interior of the spatial domain. Second, we consider Neumann boundary conditions so that no previous result can be adapted to this situation.

We consider a differential operator of order 2 \begin{document}$ n $\end{document} of the type \begin{document}$ A_n u = (-1)^n (a u^{(n)})^{(n)} $\end{document} , where \begin{document}$ a(x)>0 $\end{document} in \begin{document}$ [0, 1]\setminus\{x_0\} $\end{document} and \begin{document}$ a(x_0) = 0 $\end{document} . We show that, for any \begin{document}$ n\in{\mathbb{N}} $\end{document} , the operator \begin{document}$ -A_n $\end{document} generates a contractive analytic semigroup of angle \begin{document}$ \pi/2 $\end{document} on \begin{document}$ L^2 (0, 1) $\end{document} . Note that the domain of \begin{document}$ A_n $\end{document} depends on the type of degeneracy of \begin{document}$ a $\end{document} . Our theorems extend some previous results in [ 3 ] where \begin{document}$ n = 1 $\end{document} .

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 + λ 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 λ 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 ∞ − 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.

We consider nonlinear Neumann problems driven by the p-Laplacian plus an indefinite potential and with a superlinear reaction which need not satisfy the Ambrosetti–Rabinowitz condition. First, we prove an existence theorem, and then, under stronger conditions on the reaction, we prove a multiplicity theorem producing three nontrivial solutions. Then, we examine parametric problems with competing nonlinearities (concave and convex terms). We show that for all small values of the parameter \(\lambda >0\), the problem has five nontrivial solutions and if \(p=2\) (semilinear equation), there are six nontrivial solutions. Finally, we prove a bifurcation result describing the set of positive solutions as the parameter \(\lambda >0\) varies.

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 show Carleman estimates, observability inequalities and null controllability results for parabolic equations with non smooth coefficients degenerating at an interior point., Accepted in Memoirs of the American Mathematical Society

In the study of mathematical models, which lead to Cauchy problems for differential equations of parabolic (resp. hyperbolic) type or to an elliptic boundary value problem the following issues typically have a prominent interest: For more information please click the “Full Text” above.

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.

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 r ( x ) -Laplacian where r ( x ) is a function depending on x ∈ Ω , where Ω is a bounded smooth domain of R n . 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 ( u , v ) with both non-trivial u , v .

We give null controllability results for some degenerate parabolic equations in non divergence form on a bounded interval. In particular, the coefficient of the second order term degenerates at the extreme points of the domain. For this reason, we obtain an observability inequality for the adjoint problem. Then we prove Carleman estimates for such a problem. Finally, in a standard way, we deduce null controllability also for semilinear equations.

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.