Large time behavior for the nonlinear dissipative Boussinesq equation

IF 0.8 3区数学 Q2 MATHEMATICS

Mathematische Nachrichten Pub Date : 2025-07-13 DOI:10.1002/mana.70015

Wenhui Chen, Hiroshi Takeda

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引用次数: 0

Abstract

In this paper, we study the nonlinear dissipative Boussinesq equation in the whole space $R^{n}$ with $L^{1}$ integrable data. As our preparations, the optimal estimates as well as the optimal leading terms for the linearized model are derived by performing the Wentzel–Kramers–Brillouin (WKB) analysis and the Fourier analysis. Then, under some conditions on the power $p$ of nonlinearity, we demonstrate global (in time) existence of small data Sobolev solutions with different regularities to the nonlinear model by applying some fractional-order interpolations, where the optimal growth ( $n = 2$ ) and decay ( $n ⩾ 3$ ) estimates of solutions for large time are given. Simultaneously, we get a new large time asymptotic profile of global (in time) solutions. These results imply some influence of dispersion and dissipation on qualitative properties of solution.

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非线性耗散Boussinesq方程的大时间行为

本文研究了整个空间R n $\mathbb {R}^n$中具有L 1 $L^1$可积数据的非线性耗散Boussinesq方程。作为我们的准备工作，通过进行WKB分析和傅里叶分析，得到了线性化模型的最优估计和最优前导项。然后，在非线性幂p $p$的某些条件下，我们利用分数阶插值证明了非线性模型具有不同规律的小数据Sobolev解的全局（及时）存在性。其中给出了长时间解决方案的最佳增长（n = 2 $n=2$）和衰减（n大于或等于3 $n\geqslant 3$）估计。同时，我们得到了全局（及时）解的一个新的大时间渐近轮廓。这些结果表明，色散和耗散对溶液的定性性质有一定的影响。

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

Mathematische Nachrichten 数学-数学

CiteScore

1.50

自引率

0.00%

发文量

157

审稿时长

4-8 weeks

期刊介绍： Mathematische Nachrichten - Mathematical News publishes original papers on new results and methods that hold prospect for substantial progress in mathematics and its applications. All branches of analysis, algebra, number theory, geometry and topology, flow mechanics and theoretical aspects of stochastics are given special emphasis. Mathematische Nachrichten is indexed/abstracted in Current Contents/Physical, Chemical and Earth Sciences; Mathematical Review; Zentralblatt für Mathematik; Math Database on STN International, INSPEC; Science Citation Index