# Abdelaziz RHANDI

## Università degli Studi di Salerno

H-index: 24

Europe-Italy

## About Abdelaziz RHANDI

Abdelaziz RHANDI, With an exceptional h-index of 24 and a recent h-index of 14 (since 2020), a distinguished researcher at Università degli Studi di Salerno,

His recent articles reflect a diverse array of research interests and contributions to the field:

Well-posedness and asynchronous exponential growth of an age-weighted structured fish population model with diffusion in

On evolution equations with white-noise boundary conditions

Scaling and instantaneous blow up

Maximal regularity for vector-valued Schr\"{o}dinger operators

General kernel estimates of Schrödinger-type operators with unbounded diffusion terms

Bounds for the gradient of the transition kernel for elliptic operators with unbounded diffusion, drift and potential terms

Bi-Kolmogorov type operators and weighted Rellich’s inequalities

Parabolic equations with dynamic boundary conditions and drift terms

### Abdelaziz RHANDI Information

University | Università degli Studi di Salerno |
---|---|

Position | ___ |

Citations(all) | 1934 |

Citations(since 2020) | 728 |

Cited By | 1485 |

hIndex(all) | 24 |

hIndex(since 2020) | 14 |

i10Index(all) | 60 |

i10Index(since 2020) | 24 |

University Profile Page | Università degli Studi di Salerno |

## Top articles of Abdelaziz RHANDI

### Well-posedness and asynchronous exponential growth of an age-weighted structured fish population model with diffusion in

In the present paper, we address the asymptotic behavior of a fish population system structured in age and weight, while also incorporating spatial effects. Initially, we develop an abstract perturbation result concerning the essential spectral radius, employing the regular systems approach. Following that, we present the model in the form of a perturbed boundary problem, which involves unbounded operators on the boundary. Using time-invariant regular techniques, we construct the corresponding semigroup solution. Then, we designate an operator characteristic equation of the primary system via the radius of a bounded linear operator defined on the boundary space. Moreover, we provide a characterization of the uniform exponential stability and the asynchronous exponential growth property (AEG) by localizing the essential radius and proving the irreducibility of the perturbed semigroup. Finally, we precise the …

Authors

Samir Boujijane,Said Boulite,Mohamed Halloumi,Lahcen Maniar,Abdelaziz Rhandi

Journal

Journal of Evolution Equations

Published Date

2024/3

### On evolution equations with white-noise boundary conditions

In this paper, we delve into the study of evolution equations that exhibit white-noise boundary conditions. Our primary focus is to establish a necessary and sufficient condition for the existence of solutions, by utilizing the concept of admissible observation operators and the Yosida extension for such operators. By employing this criterion, we can derive an existence result, which directly involves the Dirichlet operator. In addition, we also introduce a Desch-Schappacher perturbation result, which proves to be instrumental in further understanding these equations. Overall, our paper presents a comprehensive analysis of evolution equations with white-noise boundary conditions, providing new insights and contributing to the existing body of knowledge in this field.

Authors

Mohamed Fkirine,Said Hadd,Abdelaziz Rhandi

Journal

Journal of Mathematical Analysis and Applications

Published Date

2024/7/1

### Scaling and instantaneous blow up

The main result is a simple proof of the Baras-Goldstein (1984) instantaneous blow up result for the heat equation with the inverse square potential. The proof relies heavily, indeed mainly, on scaling. Remarks are also given concerning the case when the underlying space ℝN is replaced by the Heisenberg group ℍN.

Authors

Gisèle Ruiz Goldstein,Jerome A Goldstein,İsmail Kömbe,Abdelaziz Rhandi

Journal

Quaestiones Mathematicae

Published Date

2024/3/29

### Maximal regularity for vector-valued Schr\"{o}dinger operators

In this paper we consider the vector-valued Schr\"{o}dinger operator , where the potential term is a matrix-valued function whose entries belong to and, for every , is a symmetric and nonnegative definite matrix, with non positive off-diagonal terms and with eigenvalues comparable each other. For this class of potential terms we obtain maximal inequality in Assuming further that the minimal eigenvalue of belongs to some reverse H\"older class of order , we obtain maximal inequality in , for in between and some .

Authors

Davide Addona,Vincenzo Leone,Luca Lorenzi,Abdelaziz Rhandi

Journal

arXiv preprint arXiv:2401.00479

Published Date

2023/12/31

### General kernel estimates of Schrödinger-type operators with unbounded diffusion terms

We first prove that the realization in the case of polynomially and exponentially growing diffusion and potential coefficients.

Authors

Loredana Caso,Markus Kunze,Marianna Porfido,Abdelaziz Rhandi

Journal

Proceedings of the Royal Society of Edinburgh Section A: Mathematics

Published Date

2023/5/19

### Bounds for the gradient of the transition kernel for elliptic operators with unbounded diffusion, drift and potential terms

Global Sobolev regularity and pointwise upper bounds for the gradient of transition densities associated with second order elliptic operators in R d was obtained in the case of bounded diffusion coefficients in [13, Section 5]. In this paper we generalize these results to the case of unbounded diffusions. Our technique is based on an approximation procedure and a De Giorgi type regularity result. We like to point out, that such an approximation procedure cannot be applied to the result in [13], since the constants in the estimates obtained in [13] depend on the infinity norm of the diffusion coefficients.

Authors

Markus Kunze,Marianna Porfido,Abdelaziz Rhandi

Journal

Discrete and Continuous Dynamical Systems-S

Published Date

2023/4/28

### Bi-Kolmogorov type operators and weighted Rellich’s inequalities

In this paper we consider the symmetric Kolmogorov operator on , where is the density of a probability measure on . Under general conditions on we prove first weighted Rellich’s inequalities and deduce that the operators L and with domain and respectively, generate analytic semigroups of contractions on . We observe that is the unique invariant measure for the semigroup generated by and as a consequence we describe the asymptotic behaviour of such semigroup and obtain some local positivity properties. As an application we study the bi-Ornstein-Uhlenbeck operator and its semigroup on .

Authors

Davide Addona,Federica Gregorio,Abdelaziz Rhandi,Cristian Tacelli

Journal

Nonlinear Differential Equations and Applications NoDEA

Published Date

2022/3

### Parabolic equations with dynamic boundary conditions and drift terms

The aim of this paper is to study the wellposedness and L2‐regularity, firstly for a linear heat equation with dynamic boundary conditions by using the approach of sesquilinear forms, and secondly for its backward adjoint equation using the Galerkin approximation and the extension semigroup to a negative Sobolev space.

Authors

A Khoutaibi,L Maniar,D Mugnolo,A Rhandi

Journal

Mathematische Nachrichten

Published Date

2022/6

### Instantaneous blowup and singular potentials on Heisenberg groups

In this paper we generalize the instantaneous blowup result from [3] and [15] to the heat equation perturbed by singular potentials on the Heisenberg group.

Authors

Gisele R Goldstein,Jerome A Goldstein,Alessia E Kogoj,Abdelaziz Rhandi,Cristian Tacelli

Journal

arXiv preprint arXiv:2204.04548

Published Date

2022/4/9

### Elliptic Equations in\mathbb R+ d with General Boundary Conditions

In this chapter, we extend the results of Chapter 11 to the case when the Dirichlet boundary conditions are replaced with more general first-order boundary conditions. More precisely, we consider the operator ( A , D ( A ) ) = Tr ( Q ∇ ) + 〈 b , ∇ 〉 + c https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429262593/03624a6d-a4e5-45fb-ba52-c2b8f08b80c3/content/math12_3.tif"/> …

Authors

Luca Lorenzi,Abdelaziz Rhandi

Published Date

2021/1/5

### Parabolic Equations in ℝ+ d with Dirichlet Boundary Conditions

Parabolic Equations in ℝ + d https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429262593/03624a6d-a4e5-45fb-ba52-c2b8f08b80c3/content/math7_3.tif"/> with Dirichlet Boundary Conditions

Authors

Luca Lorenzi,Abdelaziz Rhandi

Published Date

2021/1/5

### Elliptic Operators and Analytic Semigroups

In this chapter, taking advantage of the results proved in all the previous chapters, we show that the semigroups considered in Chapters 6 to 9 are analytic and we characterize the interpolations spaces of order α and 1 + α https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429262593/03624a6d-a4e5-45fb-ba52-c2b8f08b80c3/content/math14_1.tif"/> for every α ∈ ( 0 , 1 ) ∖ { 1 / 2 …

Authors

Luca Lorenzi,Abdelaziz Rhandi

Published Date

2021/1/5

### On vector-valued Schrödinger operators with unbounded diffusion in spaces

We prove generation results of analytic strongly continuous semigroups on () for a class of vector-valued Schrödinger operators with unbounded coefficients. We also prove Gaussian type estimates for such semigroups.

Authors

Luciana Angiuli,Luca Lorenzi,Elisabetta Mangino,Abdelaziz Rhandi

Journal

Journal of Evolution Equations

Published Date

2021/9

### Generation results for vector-valued elliptic operators with unbounded coefficients in spaces

We consider a class of vector-valued elliptic operators with unbounded coefficients, coupled up to the first order, in the Lebesgue space with . Sufficient conditions to prove generation results of an analytic -semigroup $${\varvec{T}}(t)$$, together with a characterization of the domain of its generator, are given. Some results related to the hypercontractivity and the ultraboundedness of the semigroup are also established.

Authors

Luciana Angiuli,Luca Lorenzi,Elisabetta M Mangino,Abdelaziz Rhandi

Journal

Annali di Matematica Pura ed Applicata (1923-)

Published Date

2021

### Elliptic Equations on Smooth Domains Ω

In this chapter, we analyze the boundary value problems 13.0.1 λ u − A d u = f …

Authors

Luca Lorenzi,Abdelaziz Rhandi

Published Date

2021/1/5

### Elliptic Equations in R+ d with Homogeneous Dirichlet Boundary Conditions

This chapter is the first step toward the analysis of elliptic equations on domains. Here, we consider the boundary value problem 11.0.1 { λ u − A d u = f …

Authors

Luca Lorenzi,Abdelaziz Rhandi

Published Date

2021/1/5

### Elliptic Equations in ℝ d

In this chapter, we begin the analysis of elliptic equations, considering the equation 10.0.1 λ u − A u = f https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429262593/03624a6d-a4e5-45fb-ba52-c2b8f08b80c3/content/math10_1.tif"/>

Authors

Luca Lorenzi,Abdelaziz Rhandi

Published Date

2021/1/5

### Approximate controllability of network systems.

In this paper, the rich feedback theory of regular linear systems in the Salamon-Weiss sense as well as some advanced tools in semigroup theory are used to formulate and solve control problems for network systems. In fact, we derive necessary and sufficient conditions for approximate controllability of such systems. These criteria, in some particular cases, are given by the well-known Kalman’s controllability rank condition.1. Introduction. Network systems or transport networks are mathematical models introduced to describe flow of a product from a source to a prescribed destination (sink) along the edges of a weighted oriented graph. These products can be gallons of oil flowing into pipelines, the number of telephone calls transmitted in a communication system or airplanes flying in certain regions of airspace [3]. In modeling such situations, we interpret the weight of an edge in the directed graph as a proportion that any given mass distribution should satisfy, for example, the amount of oil that can flow through a certain part of system of pipelines. To make these ideas precise, we consider a strongly connected and directed graph G=(V, E), and assume that

Authors

Yassine El Gantouh,Said Hadd,Abdelaziz Rhandi

Journal

Evolution Equations & Control Theory

Published Date

2021/9

### Parabolic Equations in ℝ d

This chapter considers the nonhomogeneous Cauchy problem, and assumes some conditions on the coefficients of the operator. It contains the optimal regularity results for the Cauchy problem. The chapter proves some interesting interior Schauder estimates satisfied by the solutions to the differential equation. It also proves that the Cauchy problem admits a unique classical solution. The chapter shows that, the more the data of the Cauchy problem are smoother, the more the solution to such a Cauchy problem is itself smooth. The results concerning existence and regularity of solutions to parabolic problems in the whole space with respect to the Holder norms are well known.

Authors

Luca Lorenzi,Abdelaziz Rhandi

Published Date

2021/1/5

### Elliptic and Parabolic Maximum Principles

This chapter focuses on the weak and strong parabolic maximum principles for second-order uniformly elliptic operators on bounded domains. Maximum principles are the fundamental tool to prove the uniqueness of solutions to parabolic Cauchy problems as well as elliptic boundary value problems. The chapter deals with solutions to the inequality and considers functions which satisfy the inequality. The main results are the weak and the strong maximum principles which hold true in both the parabolic and elliptic case. A comparison principle can be proved from the weak maximum principles. The chapter proves the uniqueness of the classical solution of the Cauchy-Dirichlet problem associated with the operator, and also considers different sets of boundary conditions, such as Neumann boundary conditions. To prove the uniqueness of the classical solution to such problems, the parabolic Hopf’s lemma plays a …

Authors

Luca Lorenzi,Abdelaziz Rhandi

Published Date

2021/1/5

### Parabolic Equations in ℝ d+ with More General Boundary Conditions

This chapter proves the last type of local Schauder estimates, up to a portion of the boundary, for solutions to parabolic equations on ∂ℝd+, which satisfy zero- or first-order boundary conditions. It also proves some other regularity results for the solution to the Cauchy problem. The chapter solves the Neumann Cauchy problem for the Laplacian, which is easier to analyze, since one can determine an explicit formula for its solution, and next using the method of continuity together with the a priori estimate. Using tools from the semigroup theory, it also solves an optimal spatial regularity result for solutions to problem.

Authors

Luca Lorenzi,Abdelaziz Rhandi

Published Date

2021/1/5

### 25th Internet Seminar on Spectral Theory for Operators and Semigroups

Along the first lecture of this Internet Seminar, we are going to consider usually a complex Banach space X, a linear operator T with domain D (T) and range Rg (T), both contained in X, and we will study the operator λ− T, where λ is a complex number. If λ is such that λ− T: D (T)→ X is bijective and λ− T has continuous inverse, then λ is said to belong to the resolvent set ρ (T) of T, otherwise λ belongs to the spectrum σ (T) of T.The main aim of Spectral Theory is the systematic analysis of the properties of T and (λ− T)− 1, through the study of the sets ρ (T) and σ (T). If X is a finite-dimensional space, then we can represent T as a matrix, and in that case spectral analysis reduces to the study of the eigenvalues of the matrix, since if λ− T does not have an inverse, then there exists x∈ X, with x= 0 such that Tx= λx. It is well known that the spectrum of a matrix contains at least one complex eigenvalue. Dealing with the infinite dimensional case, the picture gets more diversified, since, eg, the spectrum could be empty and spectral values need not to be eigenvalues. This week we start by providing the basic definitions and tools to deal with operators on Banach spaces and their spectra.

Authors

Angela A Albanese,Luca Lorenzi,Elisabetta M Mangino,Abdelaziz Rhandi

Published Date

2021/11/13

### Analytic Semigroups

This chapter focuses on the study of semigroups of bounded operators introducing analytic semigroups and analyzes their main properties. It describes analytic semigroups and explains the definition of analytic semigroups via the Dunford integral, starting from the easiest class of semigroups (which are also analytic): the uniformly continuous semigroups. The chapter introduces the sectorial operators, which are the operators which are associated to analytic semigroups. It defines the analytic semigroup associated with the sectorial operator and also describes its main properties via the Dunford integral. The chapter shows in general analytic semigroups are not strongly continuous semigroups since the operator is not required to be densely defined in the Banach space. It provides equivalent characterization of the interpolation space, which plays a relevant role in the study of abstract Cauchy problems associated …

Authors

Luca Lorenzi,Abdelaziz Rhandi

Published Date

2021/1/5

### Prelude to Parabolic Equations: The Heat Equation and the Gauss-Weierstrass Semigroup in C b (\mathbb R d)

This chapter focuses on the analysis of parabolic equations in the whole space and in bounded domains. It deals with the nonhomogeneous Cauchy problem. The chapter explains the main properties of the heat kernel and uses such results to prove that the function is the unique classical solution to the Cauchy problem. It proves two equivalent characterizations of the Holder spaces, which are given in terms of the Gauss-Weierstrass semigroup. The chapter explains the fundamental optimal Schauder estimates for solutions to the Cauchy problem. It also proves the existence and uniqueness of a classical solution to the Cauchy problem. The Gauss-Weierstrass semigroup is not strongly continuous.

Authors

Luca Lorenzi,Abdelaziz Rhandi

Published Date

2021/1/5

### Function Spaces

Both the data and the solutions for problems in PDEs are functions defined on certain domains or manifolds. In order to formulate precise theorems of existence, uniqueness, continuous dependence, etc., it is essential to specify the spaces in which these functions lie and to give a precise meaning to convergence in those spaces. This issue has led to the development of what is now considered one of the main (“core”) fields of pure mathematics, namely, functional analysis. In this and the subsequent chapter, we shall give a brief introduction to this field, with particular emphasis on those issues and concepts that are important in differential equations. This introduction is limited to what is needed in the rest of the book and is not meant as a substitute for a proper course in functional analysis.

Authors

Michael Renardy,Robert C Rogers

Journal

An Introduction to Partial Differential Equations

Published Date

2004

### Controllability of vertex delay type problems by the regular linear systems approach

In this paper, we study the well-posedness and approximate controllability of a class of network systems having delays and controls at the boundary conditions. The particularity of this work is that the network system is defined on infinite metric graphs. This fact offers many difficulties in applying the usual methods. In fact, the well-posedness of the delay network system is obtained by using a semigroup approach on product spaces which is based on the concept of feedback theory of infinite-dimensional linear systems. This technique allows us to reformulate the delay system into a free-delay distributed control system. From this transformation, we deduce necessary and sufficient conditions for the boundary approximate controllability of such systems. Furthermore, a Rank condition for the approximate controllability is also obtained. This condition coincides with the usual Kalman controllability criterion in the case of a simple transport process on a finite graph. Finally, by applying our approach to a linear Eulerian model (with airborne delays) for (ATFM), we provide a new algebraic condition for the controllability of such a model in terms of the generic rank of the so-called extended controllability matrix.

Authors

Y El Gantouh,S Hadd,A Rhandi

Journal

arXiv preprint arXiv:2110.15422

Published Date

2021/10/28

### Kernel Estimates

As it has been noticed in Chapter 5, if f is a bounded and continuous function over ℝ d and { T ( t ) } https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429262593/03624a6d-a4e5-45fb-ba52-c2b8f08b80c3/content/math15_1.tif"/> is the Gauss-Weierstrass semigroup, then, for each t > 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429262593/03624a6d-a4e5-45fb-ba52-c2b8f08b80c3/content/math15_2.tif"/> …

Authors

Luca Lorenzi,Abdelaziz Rhandi

Published Date

2021/1/5

### Strongly Continuous Semigroups

In Chapter 2 we showed that the examples of Chapter 1, which were described by ordinary differential equations, can be written as a first-order differential equation $$\dot{x}(t) = Ax(t) + Bu(t), \quad \quad y(t) = C_x(t) = Du(t),$$ where x(t) is a vector in Rn or Cn.

Authors

Birgit Jacob,Hans J Zwart,Birgit Jacob,Hans J Zwart

Journal

Linear Port-Hamiltonian Systems on Infinite-dimensional Spaces

Published Date

2012

### Parabolic Equations in Bounded Smooth Domains Ω

In this chapter, we complete the analysis of parabolic Cauchy problems with bounded coefficients, dealing with the Cauchy-Dirichlet problem 9.0.1 D t u ( t , x ) = A …

Authors

Luca Lorenzi,Abdelaziz Rhandi

Published Date

2021/1/5

### Ornstein--Uhlenbeck Semigroups on Star Graphs

We prove first existence of a classical solution to a class of parabolic problems with unbounded coefficients on metric star graphs subject to Kirchhoff-type conditions. The result is applied to the Ornstein--Uhlenbeck and the harmonic oscillator operators on metric star graphs. We give an explicit formula for the associated Ornstein--Uhlenbeck semigroup and give the unique associated invariant measure. We show that this semigroup inherits the regularity properties of the classical Ornstein--Uhlenbeck semigroup on .

Authors

Delio Mugnolo,Abdelaziz Rhandi

Journal

arXiv preprint arXiv:2109.12369

Published Date

2021/9/25

### Semigroup applications everywhere

Most dynamical systems arise from partial differential equations (PDEs) that can be represented as an abstract evolution equation on a suitable state space complemented by an initial or final condition. Thus, the system can be written as a Cauchy problem on an abstract function space with appropriate topological structures. To study the qualitative and quantitative properties of the solutions, the theory of one-parameter operator semigroups is a most powerful tool. This approach has been used by many authors and applied to quite different fields, e.g. ordinary and PDEs, nonlinear dynamical systems, control theory, functional differential and Volterra equations, mathematical physics, mathematical biology, stochastic processes. The present special issue of Philosophical Transactions includes papers on semigroups and their applications. This article is part of the theme issue ‘Semigroup applications everywhere’.

Authors

Rainer Nagel,Abdelaziz Rhandi

Published Date

2020/11/27

## Abdelaziz RHANDI FAQs

### What is Abdelaziz RHANDI's h-index at Università degli Studi di Salerno?

The h-index of Abdelaziz RHANDI has been 14 since 2020 and 24 in total.

### What are Abdelaziz RHANDI's top articles?

The articles with the titles of

Well-posedness and asynchronous exponential growth of an age-weighted structured fish population model with diffusion in

On evolution equations with white-noise boundary conditions

Scaling and instantaneous blow up

Maximal regularity for vector-valued Schr\"{o}dinger operators

General kernel estimates of Schrödinger-type operators with unbounded diffusion terms

Bounds for the gradient of the transition kernel for elliptic operators with unbounded diffusion, drift and potential terms

Bi-Kolmogorov type operators and weighted Rellich’s inequalities

Parabolic equations with dynamic boundary conditions and drift terms

...

are the top articles of Abdelaziz RHANDI at Università degli Studi di Salerno.

### What is Abdelaziz RHANDI's total number of citations?

Abdelaziz RHANDI has 1,934 citations in total.