变系数耗散波动方程的能量衰减估计

doi:10.3785/j.issn.1008-9497.2018.02.003

浙江大学学报（理学版）

2018, Vol. 45

Issue (2): 143-146 DOI: 10.3785/j.issn.1008-9497.2018.02.003

数学与计算机科学

变系数耗散波动方程的能量衰减估计

赵菁蕾¹, 吴邦²

1. 丽水学院教育学院, 浙江丽水 323000;
2. 浙江理工大学理学院, 浙江杭州 320018

Energy decay for a dissipative wave equation with variable coefficients

ZHAO Jinglei¹, WU Bang²

1. College of Education, Lishui University, Lishui 323000, Zhejiang Province, China;
2. School of Science, Zhejiang Sci-Tech University, Hangzhou 310018, China

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摘要： 研究了在（0，∞）×Rⁿ上，变系数耗散波动方程 $\100dpi \inline ${u_{tt}} - \sum\limits_{i,j = 1}^n {{\partial _{{x_i}}}} \left( {{a^{ij}}\left( x \right){\partial _{{x_j}}}u} \right) + u = 0$$ 的能量在外区域上的衰减估计，得到：若初值{u₀，u₁}属于能量空间且具有紧支集，则在Rⁿ上存在一个外区域X_m，使得对任意t ≥ 0和m>0，有 $\90dpi \inline $\int_{{X_m}} {\left( {{{\left| {{u_t}} \right|}^2} + \sum\limits_{i,j = 1}^n {{a^{ij}}\left( x \right){u_{{x_i}}}{u_{{x_j}}}} } \right)} {\rm{d}}x \le C{\left( {1 + t} \right)^{ - m}}$$ 进一步，若u₀+u₁=0，还可以得到∫_{X_m}|u|²dx ≤ C（1+t）^-m，t ≥ 0.

关键词： 能量衰减; 耗散; 变系数; 多项式衰减

Abstract: In this paper, we study the energy decay estimates to the Cauchy problem of dissipative wave equation with variable coefficients:u_tt-

∂_{x_i}(a^ij(x)∂_{x_j}u)+u_t=0. If the initial data {u₀, u₁} are compactly supported from the energy space, then there exists a exterior domain X_m⊂Rⁿ such that for large t ≥ 0, we have ∫_{X_m}(|u_t|²+

a^ij(x)u_{x_i}u_{x_j})dx ≤ C(1+t)^-m with m>0. Moreover, if u₀+u₁=0, we also have ∫_{X_m}|u|²dx ≤ C(1+t)^-m for t ≥ 0.

Key words: energy decay dissipative variable coefficients polynomial decay

收稿日期: 2017-03-01 出版日期: 2018-03-08

CLC:

O175.27

基金资助: 丽水市高层次人才项目（2016RC25）.

通讯作者: 吴邦,ORCID:http://orcid.org/0000-0002-1110-5866,E-mail:wu1109401@163.com E-mail: wu1109401@163.com

作者简介: 赵菁蕾(1975-),ORCID:http://orcid.org/0000-0001-5342-5772,女,硕士,主要从事偏微分方程研究.

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引用本文:

赵菁蕾, 吴邦. 变系数耗散波动方程的能量衰减估计[J]. 浙江大学学报（理学版）, 2018, 45(2): 143-146.

ZHAO Jinglei, WU Bang. Energy decay for a dissipative wave equation with variable coefficients. Journal of Zhejiang University (Science Edition), 2018, 45(2): 143-146.

链接本文:

https://www.zjujournals.com/sci/CN/10.3785/j.issn.1008-9497.2018.02.003 或 https://www.zjujournals.com/sci/CN/Y2018/V45/I2/143

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