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高维变区域上抛物方程的镇定

郑国杰1, 孙怡悦2, 仝荣花3   

  1. 1. 宁波财经学院数字技术与工程学院, 宁波 315175;
    2. 河南师范大学数学与信息科学学院, 新乡 453007;
    3. 确山县第一高级中学, 驻马店 463299
  • 收稿日期:2022-05-20 修回日期:2022-06-21 发布日期:2022-12-13
  • 通讯作者: 郑国杰, Email: guojiezheng@whu.edu.cn
  • 基金资助:
    河南省自然科学基金项目(202300410248),浙江省自然科学基金重点项目(LZ21A010001)资助课题.

郑国杰, 孙怡悦, 仝荣花. 高维变区域上抛物方程的镇定[J]. 系统科学与数学, 2022, 42(11): 2914-2927.

ZHENG Guojie, SUN Yiyue, TONG Ronghua. Stabilization of a Parabolic Equation on Multi-Dimensional Time-Varying Domains[J]. Journal of Systems Science and Mathematical Sciences, 2022, 42(11): 2914-2927.

Stabilization of a Parabolic Equation on Multi-Dimensional Time-Varying Domains

ZHENG Guojie1, SUN Yiyue2, TONG Ronghua3   

  1. 1. College of Digital Technology and Engineering, Ningbo University of Finance & Economics, Ningbo 315175;
    2. College of Mathematics and Information Science, Henan Normal University, Xinxiang 453007;
    3. Queshan No. 1 Middle School, Zhumadian 463299
  • Received:2022-05-20 Revised:2022-06-21 Published:2022-12-13
文章研究了高维变区域上不稳定的抛物方程的镇定问题.在研究过程中首先讨论了变区域上抛物方程的适定性.其次运用了谱分析的方法,研究了变区域上拉普拉斯算子的第一特征值的单调性, 连续性等性质;再次借助于解的正则性, 证明了 一个重要的能量等式;接着运用内部控制的技巧, 有效镇定了变区域上不稳定的抛物方程;最后给出一个例子和相应的数值实验来验证理论的有效性.
This paper aims to stabilize an unstable parabolic equation on multi-dimensional variable domains. We first investigate the well-posedness of the parabolic equation on the time-varying domains. Secondly, by using the method of spectral analysis, we study the monotonicity and continuity of the first eigenvalue of the Laplacian operator on the time-varying domains; then we obtain an important energy equation by the regularity of the solution. Next, we can stabilize the unstable parabolic equation with the help of internal control. Finally, an example and its numerical results are given to verify the validity of the proposed theory.

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