Wyniki wyszukiwania - BazTech

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

|

2023

|

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

Global existence and blow-up phenomenon for a quasilinear viscoelastic equation with strong damping and source terms

Di Huafei, Song Zefang

Opuscula Mathematica

|

2022

|

Vol. 42, no. 2

119--155

EN

Considered herein is the global existence and non-global existence of the initial-boundary value problem for a quasilinear viscoelastic equation with strong damping and source terms. Firstly, we introduce a family of potential wells and give the invariance of some sets, which are essential to derive the main results. Secondly, we establish the existence of global weak solutions under the low initial energy and critical initial energy by the combination of the Galerkin approximation and improved potential well method involving with t. Thirdly, we obtain the finite time blow-up result for certain solutions with the non-positive initial energy and positive initial energy, and then give the upper bound for the blow-up time T∗. Especially, the threshold result between global existence and non-global existence is given under some certain conditions. Finally, a lower bound for the life span T∗ is derived by the means of integro-differential inequality techniques.

3

Blow-up of solutions for a Kirchhoff type equation with variable-exponent nonlinearities

Shahrouzi Mohammad, Kargarfard Firoozeh

Journal of Applied Analysis

|

2021

|

Vol. 27, nr 1

97--105

EN

This paper deals with a Kirchhoff type equation with variable exponent nonlinearities, subject to a nonlinear boundary condition.Under appropriate conditions and regarding arbitrary positive initial energy, it is proved that solutions blow up in a finite time.Moreover, we obtain the upper bound estimate of the blowup time.

4

Global existence and blow up of solution for semi-linear hyperbolic equation with the product of logarithmic and power-type nonlinearity

Lian Wei, Ahmed Md Salik, Xu Runzhang

Opuscula Mathematica

|

2020

|

Vol. 40, no. 1

111--130

EN

In this paper we consider the semilinear wave equation with the multiplication of logarithmic and polynomial nonlinearities. We establish the global existence and finite time blow up of solutions at three different energy levels (E(0) < d, E(0) = d and E(0) > 0) using potential well method. The results in this article shed some light on using potential wells to classify the solutions of the semilinear wave equation with the product of polynomial and logarithmic nonlinearity.

5

Transient flow in gas networks: Traveling waves

Gugat M., Wintergerst D.

International Journal of Applied Mathematics and Computer Science

|

2018

|

Vol. 28, no. 2

341--348

EN

In the context of gas transportation, analytical solutions are helpful for the understanding of the underlying dynamics governed by a system of partial differential equations. We derive traveling wave solutions for the one-dimensional isothermal Euler equations, where an affine linear compressibility factor is used to describe the correlation between density and pressure. We show that, for this compressibility factor model, traveling wave solutions blow up in finite time. We then extend our analysis to networks under appropriate coupling conditions and derive compatibility conditions for the network nodes such that the traveling waves can travel through the nodes. Our result allows us to obtain an explicit solution for a certain optimal boundary control problem for the pipeline flow.

6

Nonexistence of global solutions to a class of nonlinear differential inequalities and application to hyperbolic-elliptic problems

Alaa N., Guedda M.

Journal of Applied Analysis

|

2003

|

Vol. 9, nr 1

103-121

EN

We consider the problem [wzór] posed in Ω x (0,+∞). Here Ω ⊂ Rn is a an open smooth bounded domain and φ is like [wzór] and ε = š1. We prove, in certain conditions on f and φ that there is absence of global solutions. The method of proof relies on a simple analysis of the ordinary inequality of the type w'' + δw' ≥ αw + βwp. It is also shown that a global positive solution, when it exists, must decay at least exponentially.