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Alexandre Thorel

Publications and source records attributed to Alexandre Thorel.

12 recordsLinked to original sources
arXiv2026

Analytic semigroup generated by the dispersal process of a sylvatic transmission model of Chagas disease

In this work, we develop a new biological transmission model for Chagas disease. This model, set in two juxtaposed habitats with skew Brownian motion conditions at the interface, is composed of two reaction--diffusion equations and takes into account the sylvatic transmission. We write it as an abstract perturbed Cauchy problem using operator theory. Then, we show that the main operator, which models the dispersal process, generates an analytic semigroup in an adequate Banach space.

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arXiv2025

Bounded imaginary powers of generalized diffusion operators

In this paper, we investigate the boundedness of the imaginary powers of four generalized diffusion operators. This key property, which implies the maximal regularity property, allows us to solve both the linear and semilinear Cauchy problems associated with each operator. Our approach relies on semigroup theory, functional calculus, operator sum theory and R-boundedness techniques to establish the boundedness of the imaginary powers of generalized diffusion operators. We then apply the Dore-Venni theorem to solve the linear problem, obtaining a unique solution with maximal regularity. Finally, we tackle the semilinear problem and prove the existence of a unique global solution.

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arXiv2024

Solvability of a transmission problem in $L^p$-spaces with generalized diffusion equation

We study a transmission problem, in population dynamics, between two juxtaposed habitats. In each habitat, we consider a generalized diffusion equation composed by the Laplace operator and a biharmonic term. We consider that the coefficients in front of each term could be negative or null. Using semigroups theory and functional calculus, we give some relation between coefficients to obtain the existence and the uniqueness of the classical solution in $L^p$-spaces.

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arXiv2024

Solvability of a fourth order elliptic problem in a bounded sector, part II

After different variables and functions changes, the generalized dispersal problem, recalled in (1) below and considered in part I, see Labbas, Maingot and Thorel [14], leads us to consider, to study and to invert the sum of linear operators (4) below in a suitable Banach space by using two strategies: namely the theory of sums of operators in Banach spaces as developed by Da Prato-Grisvard [4] and successfully improved by Dore-Venni [5].

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arXiv2023

Solvability of a fourth order elliptic problem in a bounded sector, part I

The purpose of this article (composed of two parts) is the study of the generalized dispersal operator of a reaction-diffusion equation in $L^p$-spaces set in the finite conical domain $S_{\omega,\rho}$ of angle $\omega>0$ and radius $\rho$ > 0 in $\mathbb{R}^2$. This first part is devoted to the behavior of the solution near the top of the cone which is completely described in the weighted Sobolev space $W^{4,p}_{3-\frac{1}{p}}(S_{\omega,\rho_0})$, $\rho_0 \leqslant \rho$, see Theorem 2.2.

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arXiv2022

Generalized diffusion problems in a conical domain, part I

The purpose of this article (composed of two parts) is the study of the generalized dispersal operator of a reaction-diffusion equation in $L^p$-spaces set in the finite conical domain $S_{\omega,\rho}$ of angle $\omega>0$ and radius $\rho>0$ in $\mathbb{R}^2$. This first part is devoted to the behaviour of the solution near the top of the cone which is completely described in the weighted Sobolev space $W^{4,p}_{3-\frac{1}{p}}(S_{\omega,\rho})$, see Theorem 2.2.

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arXiv2022

Generalized diffusion problems in a conical domain, part II

After different variables and functions changes, the generalized dispersal problem, recalled in (1) below and considered in part I, see [14], leads us to invert a sum of linear operators in a suitable Banach space, see (2) below. The essential result of this second part lies in the complete study of this sum using the two well-known strategies: the one of Da Prato-Grisvard [4] and the one of Dore-Venni [6].

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arXiv2021

A biharmonic transmission problem in Lp-spaces

In this work we study, by a semigroup approach, a transmission problem based on biharmonic equations with boundary and transmission conditions, in two juxtaposed habitats. We give a result of existence and uniqueness of the classical solution in L p-spaces, for p $\in$ (1, +$\infty$), using analytic semigroups and operators sum theory in Banach spaces. To this end, we invert explicitly the determinant operator of the transmission system in L p-spaces using the E $\infty$-calculus and the Dore-Venni sums theory.

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arXiv2021

Elliptic differential-operator with an abstract Robin boundary condition containing two spectral parameters, study in a non commutative framework

We study the solvability of boundary-value problems for differential-operator equations of the second order in L p (0, 1; X), with 1 < p < +$\infty$, X being a UMD complex Banach space. The originality of this work lies in the fact that we have considered the case when spectral complex parameters appear in the equation and in the abstract Robin boundary condition illustrated by some unbounded operator non commuting with the one used in the equation. Existence, uniqueness, representation formula, maximal regularity of the solution, sharp estimates and generation of strongly continuous analytic semigroup are proved. Many concrete applications are given for which our theory applies. This work gives news considerations with respect to all those studied by the authors in [7] and is a continuation, in some sense, of the results in [1] studied in Hilbertian spaces.

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arXiv2021

An integrated semigroup approach for age structured equations with diffusion and non-homogeneous boundary conditions

In this work, we consider a linear age-structured problem with diffusion and non-homogeneous boundary conditions both for the age and the space variables. We handle this linear problem by re-writing it as a non-densely defined abstract Cauchy problem. To that aim we develop a new result on the closedness of a commutative sum of two non-densely defined operators by using the theory of integrated semigroups. As an application of this abstract result, we are able to associate a suitable integrated semigroup to some age-structured problem with spatial diffusion and equipped with non-homogeneous boundary conditions. This integrated semigroup is characterized by the description of its infinitesimal generator. Further applications of our abstract result are also given to the commutative sum of two almost sectorial operators, for which we derive a closedness results.

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