Opuscula Mathematica
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ISSN 1232-9274
e-ISSN: 2300-6919
Issue Date
2023
Volume
Vol. 43
Number
No. 6
Description
Journal Volume
Opuscula Mathematica
Vol. 43 (2023)
Projects
Pages
Articles
Nonexistence of global solutions for a nonlinear parabolic equation with a forcing term
(Wydawnictwa AGH, 2023) Alshehri, Aisha; Aljaber, Noha; Altamimi, Haya; Alessa, Rasha; Majdoub, Mohamed
The purpose of this work is to analyze the blow-up of solutions of a nonlinear parabolic equation with a forcing term depending on both time and space variables $u_t-\Delta u=|x|^{\alpha} |u|^{p}+{\mathtt a}(t)\,{\mathbf w}(x)\quad\text{for }(t,x)\in(0,\infty)\times \mathbb{R}^{N},$ where $\alpha \in \mathbb{R}$, $p \gt 1$, and ${\mathtt a}(t)$ as well as ${\mathbf w}(x)$ are suitable given functions. We generalize and somehow improve earlier existing works by considering a wide class of forcing terms that includes the most common investigated example $t^\sigma\,{\mathbf w}(x)$ as a particular case. Using the test function method and some differential inequalities, we obtain sufficient criteria for the nonexistence of global weak solutions. This criterion mainly depends on the value of the limit $\lim_{t\to\infty} \frac{1}{t}\,\int_0^t\,{\mathtt a}(s)\,ds$. The main novelty lies in our treatment of the nonstandard condition on the forcing term.
Regularity and existence of solutions to parabolic equations with nonstandard p(x,t),q(x,t)-growth conditions
(Wydawnictwa AGH, 2023) Bahja, Hamid El
We study the Cauchy-Dirichlet problem for a class of nonlinear parabolic equations driven by nonstandard $p(x,t),q(x,t)$-growth condition. We prove theorems of existence and uniqueness of weak solutions in suitable Orlicz-Sobolev spaces, derive global and local in time $L^{\infty}$ bounds for the weak solutions.
Oscillation conditions for difference equations with several variable delays
(Wydawnictwa AGH, 2023) El-Matary, Bassant M.; El-Morshedy, Hassan A.; Benekas, Vasileios; Stavroulakis, Ioannis P.
A technique is developed to establish a new oscillation criterion for a first-order linear difference equation with several delays and non-negative coefficients. Our result improves recent oscillation criteria and covers the cases of monotone and non-monotone delays. Moreover, the paper is concluded with an illustrative example to show the applicability and strength of our result.
On the concept of generalization of I-density points
(Wydawnictwa AGH, 2023) Hejduk, Jacek; Wiertelak, Renata
This paper deals with essential generalization of $\mathcal{I}$-density points and I-density topology. In particular, there is an example showing that this generalization of $\mathcal{I}$-density point yields the stronger concept of density point than the notion of $\mathcal{I}(\mathcal{J})$-density. Some properties of topologies generated by operators related to this essential generalization of density points are provided.
On minimum intersections of certain secondary dominating sets in graphs
(Wydawnictwa AGH, 2023) Kosiorowska, Anna; Michalski, Adrian; Włoch, Iwona
In this paper we consider secondary dominating sets, also named as $(1,k)$-dominating sets, introduced by Hedetniemi et al. in 2008. In particular, we study intersections of the $(1,1)$-dominating sets and proper $(1,2)$-dominating sets. We introduce $(1,\overline{2})$-intersection index as the minimum possible cardinality of such intersection and determine its value for some classes of graphs.