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1

Nonexistence of global solutions for a nonlinear parabolic equation with a forcing term

Alshehri Aisha, Aljaber Noha, Altamimi Haya, Alessa Rasha, Majdoub Mohamed

Opuscula Mathematica

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2023

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Vol. 43, no. 6

741--758

EN

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 ut − Δu = |x|α|u|p + a(t)w(x) for (t, x) ∈ (0,∞) × RN, where α ∈ R, p > 1, and a(t) as well as 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 tσ 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 [formula]. The main novelty lies in our treatment of the nonstandard condition on the forcing term.

2

On three methods for bounding the rate of convergence for some continuous-time Markov chains

Zeifman Alexander, Satin Yacov, Kryukova Anastasia, Razumchik Rostislav, Kiseleva Ksenia, Shilova Galina

International Journal of Applied Mathematics and Computer Science

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2020

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Vol. 30, no. 2

251--266

EN

Consideration is given to three different analytical methods for the computation of upper bounds for the rate of convergence to the limiting regime of one specific class of (in)homogeneous continuous-time Markov chains. This class is particularly well suited to describe evolutions of the total number of customers in (in)homogeneous M/M/S queueing systems with possibly state-dependent arrival and service intensities, batch arrivals and services. One of the methods is based on the logarithmic norm of a linear operator function; the other two rely on Lyapunov functions and differential inequalities, respectively. Less restrictive conditions (compared with those known from the literature) under which the methods are applicable are being formulated. Two numerical examples are given. It is also shown that, for homogeneous birth-death Markov processes defined on a finite state space with all transition rates being positive, all methods yield the same sharp upper bound.

3

Concavity of solutions of a 2n-th order problem with symmetry

Twaty A. A., Eloe P. W.

Opuscula Mathematica

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2013

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Vol. 33, no. 4

603--613

EN

In this article we apply an extension of a Leggett-Williams type fixed point theorem to a two-point boundary value problem for a 2n-th order ordinary differential equation. The fixed point theorem employs concave and convex functionals defined on a cone in a Banach space. Inequalities that extend the notion of concavity to 2n-th order differential inequalities are derived and employed to provide the necessary estimates. Symmetry is employed in the construction of the appropriate Banach space.

4

Second order differential inequalities via aggregation

Herzog G., Lemmert R.

Demonstratio Mathematica

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2007

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Vol. 40, nr 2

347-364

EN

Let E be a Banach space ordered by a solid and normal cone. We introduce a polynorm with respect to a given selection of positive pairwise disjoint vectors p1, . . . , pm, and derive monotonicity properties of solutions of second order differential inequalities under one-sided matrix Lipschitz conditions.

5

Numerical method of lines for first order partial differential equations with deviated variables

Czernous W.

Demonstratio Mathematica

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2005

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Vol. 38, nr 1

239--254

EN

Classical solutions of nonlinear initial boundary value problems are approximated in the paper by solutions of suitable quasilinear differential difference systems. The proof of the stability of the method of lines is based on a comparison technique with nonlinear estimates of the Perron type. Numerical examples are given.

6

On-sided estimates for quasimonotone systems of boundary value problem

Herzog G.

Demonstratio Mathematica

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2004

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Vol. 37, nr 4

847--856

EN

We prove existence and uniqueness theorems for Dirichlet boundary value problems of the form u" + f(t,u) = 0, u(0) = uo, u(1) = ui in ordered finite dimensional Banach spaces, involving one-sided estimates and quasimonotonicity.

7

Equipower and equichordal extension

Stępnicki R.

Demonstratio Mathematica

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2004

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Vol. 37, nr 4

925--938

EN

In this paper we consider the possibility of extension of a concave function f : [0, a] -> [0, +oo) to equipower convex curve or equichordal convex curve with axis of symmetry. The extension is possible if and only if f satisfies a differential inequality of the second degree.