Opuscula Mathematica
Loading...
ISSN 1232-9274
e-ISSN: 2300-6919
Issue Date
2023
Volume
Vol. 43
Number
No. 1
Description
Journal Volume
Opuscula Mathematica
Vol. 43 (2023)
Projects
Pages
Articles
On the blowing up solutions of the 4-d general q-Kuramoto-Sivashinsky equation with exponentially »dominated« nonlinearity and singular weight
(Wydawnictwa AGH, 2023) Baraket, Sami; Mahdaoui, Safia; Ouni, Taieb
Let $\Omega$ be a bounded domain in $\mathbb{R}^{4}$ with smooth boundary and let $x^{1},x^{2},\dots,x^{m}$ be $m$-points in $\Omega$. We are concerned with the problem $\Delta^{2} u - H(x,u,D^{k}u) = \rho^{4}\prod_{i=1}^{n}|x-p_{i}|^{4\alpha_{i}}f(x)g(u),$ where the principal term is the bi-Laplacian operator, $H(x,u,D^{k}u)$ is a functional which grows with respect to $Du$ at most like $|Du|^{q}$, $1 \leq q \leq 4$, $f:\Omega\to [0,+\infty[$ is a smooth function satisfying $f(p_{i}) \gt 0$ for any $i = 1,\ldots, n$, $\alpha_{i}$ are positives numbers and $g:\mathbb{R} \to [0,+\infty[$ satisfy $|g(u)|\leq ce^{u}$. In this paper, we give sufficient conditions for existence of a family of positive weak solutions $(u_\rho)_{\rho\gt 0}$ in $\Omega$ under Navier boundary conditions $u=\Delta u =0$ on $\partial \Omega$. The solutions we constructed are singular as the parameters $ho$ tends to 0, when the set of concentration $S=\{x^{1},\ldots,x^{m}\}\subset\Omega$ and the set $\Lambda :=\{p_{1},\ldots, p_{n}\}\subset\Omega$ are not necessarily disjoint. The proof is mainly based on nonlinear domain decomposition method.
Singular elliptic problems with Dirichlet or mixed Dirichlet-Neumann non-homogeneous boundary conditions
(Wydawnictwa AGH, 2023) Godoy, Tomas
Let $\Omega$ be a $C^2$ bounded domain in $\mathbb{R}^{n}$ such that $\partial\Omega=\Gamma_{1}\cup\Gamma_{2}$, where $\Gamma_1$ and $\Gamma_2$ are disjoint closed subsets of $\partial \Omega$, and consider the problem $-\Delta u=g(\cdot,u)$ in $\Omega$, $u=\tau$ on $\Gamma_1$, $\frac{\partial u}{\partial\nu}=\eta$ on $\Gamma_2$, where $0\leq\tau\in W^{\frac{1}{2},2}(\Gamma_{1})$, $\eta\in(H_{0,\Gamma_{1}}^{1}(\Omega))^{\prime}$, and $g:\Omega \times(0,\infty)\rightarrow\mathbb{R}$ is a nonnegative Carathéodory function. Under suitable assumptions on $g$ and $\eta$ we prove the existence and uniqueness of a positive weak solution of this problem. Our assumptions allow $g$ to be singular at $s=0$ and also at $x \in S$ for some suitable subsets $S\subset\overline{\Omega}$. The Dirichlet problem $-\Delta u=g(\cdot,u)$ in $\Omega$, $u=\sigma$ on $\partial \Omega$ is also studied in the case when $0\leq\sigma\in W^{\frac{1}{2},2}(\Omega)$.
Existence of positive radial solutions to a p-Laplacian Kirchhoff type problem on the exterior of a ball
(Wydawnictwa AGH, 2023) Graef, John R.; Hebboul, Doudja; Moussaoui, Toufik
In this paper the authors study the existence of positive radial solutions to the Kirchhoff type problem involving the $p$-Laplacian $-\Big(a+b\int_{\Omega_e}|\nabla u|^p dx\Big)\Delta_p u=\lambda f\left(|x|,u\right),\ x\in \Omega_e,\quad u=0\ \text{on} \ \partial\Omega_e,$ where $\lambda \gt 0$ is a parameter, $\Omega_e = \lbrace x\in\mathbb{R}^N : |x|\gt r_0\rbrace$, $r_{0} \gt 0$, $N \gt p \gt 1$, $\Delta_{p}$ is the $p$-Laplacian operator, and $f\in C(\left[ r_0, +\infty\right)\times\left[0,+\infty\right),\mathbb{R})$ is a non-decreasing function with respect to its second variable. By using the Mountain Pass Theorem, they prove the existence of positive radial solutions for small values of $\lambda$.
Nonoscillation of damped linear differential equations with a proportional derivative controller and its application to Whittaker-Hill-type and Mathieu-type equations
(Wydawnictwa AGH, 2023) Ishibashi, Kazuki
The proportional derivative (PD) controller of a differential operator is commonly referred to as the conformable derivative. In this paper, we derive a nonoscillation theorem for damped linear differential equations with a differential operator using the conformable derivative of control theory. The proof of the nonoscillation theorem utilizes the Riccati inequality corresponding to the equation considered. The provided nonoscillation theorem gives the nonoscillatory condition for a damped Euler-type differential equation with a PD controller. Moreover, the nonoscillation of the equation with a PD controller that can generalize Whittaker-Hill-type equations is also considered in this paper. The Whittaker-Hill-type equation considered in this study also includes the Mathieu-type equation. As a subtopic of this work, we consider the nonoscillation of Mathieu-type equations with a PD controller while making full use of numerical simulations.
New results on imbalance graphic graphs
(Wydawnictwa AGH, 2023) Kozerenko, Sergìj Oleksandrovič; Serdûk, Andrìj
An edge imbalance provides a local measure of how irregular a given graph is. In this paper, we study graphs with graphic imbalance sequences. We give a new proof of imbalance graphicness for trees and use the new idea to prove that the same holds for unicyclic graphs. We then show that antiregular graphs are imbalance graphic and consider the join operation on graphs as well as the double graph operation. Our main results are concerning imbalance graphicness of three classes of block graphs: block graphs having all cut vertices in a single block, block graphs in which the subgraph induced by the cut vertices is either a star or a path. In the end, we discuss open questions and conjectures regarding imbalance graphic graphs.