-
À paraître

Avancées en Mathématiques Pures et Appliquées

Articles parus

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.

Fractal approximants developed through iterated function systems (IFS) prove more versatile than classical approximants. In this paper, we introduce a new class of fractal approximants using the suitable bounded linear operators defined on the space C(I) of continuous functions and concept of $$$\alpha$$$-fractal functions. The convergence of the proposed fractal approximants towards the original continuous function does not need any condition on the scaling factors. The fractal approximants proposed in this paper possess fractality and convergence simultaneously. Without imposing any condition on the scaling vector, we establish the constrained approximation by the proposed fractal approximants. Existence of Schauder basis of fractal polynomials for the space of continuous functions C(I) is investigated. Using the proposed class of fractal approximants, we develop complex fractal approximants for representation of the square integrable complex-valued functions defined on a real compact interval.

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.

In this paper we consider a nonlinear system of two coupled viscoelastic equations, prove the well posedness, and investigate the asymptotic behaviour of this system. We use minimal and general conditions on the relaxation functions and establish explicit energy decay formula which gives the best decay rates expected under this level of generality. Our new result generalizes the earlier related results in the literature.

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 $$$\alpha_\star = \frac{4-2b}{d}$$$.

We generalize Wiener amalgam spaces by using Dunkl translation instead of the classical one, and we give some relationship between these spaces, Dunkl-Lebesgue spaces and Dunkl-Morrey spaces. We prove that the Hardy-Litlewood maximal function associated with the Dunkl operators is bounded on these generalized Dunkl-Morrey spaces.

$$$\mbox{In this paper we study the nonexistence of finite Morse index solutions of the following Neumann boundary value problems}\\ {(Eq.H)} \begin{cases} -\Delta u = (u^{+})^{p} \;\; \text{in $ \mathbb{R}_+^N$}, \\ \frac{\partial u}{\partial x_{N}}=0 \quad\quad\;\; \text{ on $ \partial\mathbb{R}_+^N$}, \\ u \in C^2(\overline{\mathbb{R}_+^N}) \mbox{ and sign-changing, }\\u^+ \mbox{ is bounded and } i(u)<\infty,\end{cases}\\ \mbox{or}\\ {(Eq.H')}\begin{cases}-\Delta u = |u|^{p-1}u \text{ in $ \mathbb{R}_+^N$}, \\ \frac{\partial u}{\partial x_{N}}=0 \;\;\;\;\;\;\;\;\quad\text{ on $ \partial\mathbb{R}_+^N$}, \\ u \in C^2(\overline{\mathbb{R}_+^N}),\\ u \mbox{ is bounded and } i(u) < \infty.\end{cases}\\ \mbox{ As a consequence, we establish the relevant Bahri-Lions's }L^\infty\mbox{-estimate [3] via the boundedness of Morse index of solutions to}\\ \begin{equation}\label{1.1} \left\{\begin{array}{lll} -\Delta u=f(x,u) &\text{in $ \Omega,$}\\ \frac{\partial u}{\partial \nu}=0 &\text{on $\partial \Omega,$} \end{array} \right. \end{equation}\\ \mbox{where} f \mbox{ has an asymptotical behavior at in-nity} \mbox{which is not necessarily the same at} \pm\infty. \mbox{Our results complete previous Liouville}\\ \mbox{ type theorems and } L^\infty\mbox{-bounds via Morse index obtained in [3, 6, 13, 16, 12, 21].}$$$

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.

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.