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## Articles parus

[Forthcoming] Contrôle optimal d’une diffusion fractionnée – Problème de Sturm-Liouville sur un graphe étoile

This paper is devoted to parabolic fractional boundary value problems involving fractional derivative of Sturm-Liouville type. We investigate the existence and uniqueness results on an open bounded real interval, prove the existence of solutions to a quadratic boundary optimal control problem and provide a characterization via optimality system. We then investigate the analogous problems for a parabolic fractional Sturm-Liouville problem on a star graph with mixed Dirichlet and Neumann boundary controls.

[Forthcoming] Opérateurs de composition pondérés symétriques complexes sur un espace de Hardy à poids

In this paper various conditions under which a weighted composition operator $W_{\psi,\phi}$ on the weighted Hardy space $H^2(\beta)$ becomes complex symmetric with respect to some special conjugation have been explored. We also investigate some important properties of the complex symmetric operator $W_{\psi,\phi}$ such as hermiticity and isometry.

[Forthcoming] Théorie de diffusion dans les espaces L2 pondérés pour une classe de l’équation de Schrödinger non-linéaire inhomogène défocalisée

In this paper, we consider the following inhomogeneous nonlinear Schrödinger equation (INLS)
$i\partial_t u + \Delta u + \mu$ |$x$|$^{-b}$|$u$|$^\alpha u = 0, \quad (t,x)\in ℝ \times ℝ^d$
with $b, \alpha$ > 0. First, we revisit the local well-posedness in $H^1(ℝ^d)$ for (INLS) of Guzmán [Nonlinear Anal. Real World Appl. 37 (2017), 249-286] and give an improvement of this result in the two and three spatial dimensional cases. Second, we study the decay of global solutions for the defocusing (INLS), i.e. $\mu=-1$ when 0 < $\alpha$ < $\alpha^\star$ where $\alpha^\star = \frac{4-2b}{d-2}$ for $d\geq 3$, and $\alpha^\star = \infty$ for $d=1, 2$
by assuming that the initial data belongs to the weighted $L^2$ space $\Sigma =\{u \in H^1(ℝ^d) :$ |$x$|$u \in L^2(ℝ^d) \}$. Finally, we combine the local theory and the decaying property to show the scattering in $\Sigma$ for the defocusing (INLS) in the case $\alpha_\star$ < $\alpha$ < $\alpha^\star$, where $\alpha_\star = \frac{4-2b}{d}$.

[Forthcoming] Déficience D’une Onde Viscoélastique Non Linéaire : Équation Avec Dissipation Aux Limites

In this work we establish a general decay rate for a nonlinear viscoelastic wave equation with boundary dissipation where the relaxation function satisfies $g^{\prime }\left( t\right) \leq -\xi \left( t\right) g^{p} % \left( t\right) , t\geq 0, 1\leq p\leq \frac{3}{2}.$ This work generalizes and improves earlier results in the literature.

[Forthcoming] Multiplicité de solutions pour un problème non homogène impliquant un potentiel dans les espaces d’Orlicz-Sobolev

This paper is devoted to the study of the nonhomogeneous problem
$-div (a(|\nabla u|)\nabla u)+a(| u|)u=\lambda V(x)|u|^{m(x)-2}u-\mu g(x,u) \mbox{ in} \ \Omega, \ u=0 \mbox{ on} \ \partial\Omega ,$ where $\Omega$ is a bounded smooth domain in $\mathbb{R}^N,\lambda, \mu$ are positive real numbers, $V(x)$ is a potential, $m: \overline{ \Omega} \to (1, \infty)$ is a continuous function, $a$ is mapping such that $\varphi(|t|)t$ is increasing homeomorphism from ℝ to ℝ and $g: \overline{\Omega}\times ℝ \to ℝ$ is a continuous function. We establish there main results with various assumptions, the first one asserts that any $\lambda$0> is an eigenvalue of our problem. The second Theorem states the existence of a constant $\lambda^{*}$ such that every $\lambda \in (0,\lambda^{*})$ is an eigenvalue of the problem. While the third Theorem claims the existence of a constant $\lambda^{**}$ such that every $\lambda \in [\lambda^{**},\infty)$ is an eigenvalue of the problem. Our approach relies on adequate variational methods in Orlicz-Sobolev spaces.

### Autres numéros :

2020

Volume 20- 11

Numéro 1 (Mai 2020)
Numéro 2 (Septembre 2020)

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2021

Volume 21- 12

Numéro 1 (Janvier 2021)
Numéro 2 (Mai 2021)