The Exponential Growth of Solution, Upper and Lower Bounds for the Blow-Up Time for a Viscoelastic Wave Equation with Variable- Exponent Nonlinearities

Q3 Mathematics

WSEAS Transactions on Mathematics Pub Date : 2023-06-02 DOI:10.37394/23206.2023.22.51

Soufiane Benkouider, Abita Rahmoune

引用次数: 0

Abstract

This paper aims to study the model of a nonlinear viscoelastic wave equation with damping and source terms involving variable-exponent nonlinearities. First, we prove that the energy grows exponentially, and thus in 𝐿p2 and 𝐿p1 norms. For the case 2 ≤ 𝑘(. ) < 𝑝(. ), we reach the exponential growth result of a blowup in finite time with positive initial energy and get the upper bound for the blow-up time. For the case 𝑘(. ) = 2, we use the concavity method to show a finite time blow-up result and get the upper bound for the blow-up time. Furthermore, for the case 𝑘(. ) ≥ 2, under some conditions on the data, we give a lower bound for the blow-up time when the blow-up occurs.

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变指数非线性粘弹性波动方程解的指数增长及其爆破时间的上下限

本文的目的是研究具有阻尼和源项的非线性粘弹性波动方程的变指数非线性模型。首先，我们证明了能量呈指数增长，因此在𝐿p2和𝐿p1范数中。对于2≤𝑘(。)<𝑝(.)得到了有限时间内具有正初始能量的爆破的指数增长结果，并得到了爆破时间的上界。对于这种情况𝑘(。)= 2时，利用凹度法给出了一个有限时间爆破结果，并得到了爆破时间的上界。此外，对于𝑘(.)≥2，在数据的某些条件下，我们给出了爆炸发生时爆炸时间的下界。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

WSEAS Transactions on Mathematics Mathematics-Discrete Mathematics and Combinatorics

CiteScore

1.30

自引率

0.00%

发文量

期刊介绍： WSEAS Transactions on Mathematics publishes original research papers relating to applied and theoretical mathematics. We aim to bring important work to a wide international audience and therefore only publish papers of exceptional scientific value that advance our understanding of these particular areas. The research presented must transcend the limits of case studies, while both experimental and theoretical studies are accepted. It is a multi-disciplinary journal and therefore its content mirrors the diverse interests and approaches of scholars involved with linear algebra, numerical analysis, differential equations, statistics and related areas. We also welcome scholarly contributions from officials with government agencies, international agencies, and non-governmental organizations.