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Advances in Pure and Applied Mathematics

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[Forthcoming] On the control of a nonlinear system of viscoelastic equations

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.

[Forthcoming] Scattering theory in weighted L2 space for a class of the defocusing inhomogeneous nonlinear Schrödinger equation

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] Invariant regions and existence of global solutions to a generalized m-component reaction-diffusion system with tridiagonal symmetric Toeplitz diffusion matrix

The aim of this paper is to construct invariant regions of a generalized m-component reaction-diffusion system with tridiagonal symmetric Toeplitz diffusion matrix and nonhomogeneous boundary conditions and polynomial growth for the nonlinear reaction terms. Using the eigenvalues and eigenvectors of the diffusion matrix and the parabolicity conditions. So we prove the global existence of solutions using Lyapunov functional.

[Forthcoming] Maximal operator in Dunkl-Fofana spaces

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.

[Forthcoming] A priori estimates for super-linear elliptic equation: the Neumann boundary value problem

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

[Forthcoming] General Decay Of A Nonlinear Viscoelastic Wave: Equation With Boundary Dissipation

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] Initial value problem for the nonconservative zero-pressure gas dynamics system

In this article, we study initial value problem for the zero-pressure gas dynamics system in non conservative form and the associated adhesion approximation. We use adhesion approximation and modi-ed adhesion approximation in the construction of weak asymptotic solution. First we prove a general existence result for the adhesion model for the initial velocity component in $$$H^s \mbox{ for } s$$$ > $$$ \frac{n}{2} + 1$$$ and the initial data for the density component being a $$$C^1$$$ function. Using this, we construct weak asymptotic solution for the system with initial velocity in $$$L^2 \cap L^{\infty}$$$ and the initial density being a bounded Borel measure. Then we make a detailed analysis of the explicit formula for the weak asymptotic solution and generalized solution for the plane-wave type initial data.

[Forthcoming] Multiplicity of solutions for a nonhomogeneous problem involving a potential in Orlicz-Sobolev spaces

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.

[Forthcoming] On the existence of solutions of a nonlocal biharmonic problem

This paper is concerned with the existence of an eigenvalue for a p(x)-biharmonic Kirchhoff problem with Navier boundary condition. Under some suitable conditions, we establish that any λ > 0 is an eigenvalue . The proofs combine variational methods with energy estimates. The main results of this paper improve and generalize the previous one introduced by Kefi and Rădulescu (Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 29 (2018), 439-463).

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Volume 20- 11

Issue 1 (May 2020)
Issue 2 (September 2020)

Forthcoming papers