Strong damping wave equations defined by a class of self-similar measures with overlaps,

Wei Tang Hunan First Normal University Zhiyong Wang Hunan First Normal University

Analysis of PDEs mathscidoc:2108.03001

2021.3
The weak well-posedness of strong damping wave equations de fined by fractal Laplacians is proved by using Galerkin method. These fractal Laplacians are de fined by self-similar measures with overlaps, such as the well-known infi nite Bernoulli convolution associated with the golden ratio, the three-fold convolution of the Cantor measure, and a class of self-similar measures that we call essentially of finite type. In general, the structure of self-similar measures with overlap are complicated and intractable. However, some important information about the structure of the three measures above can be obtained. We make use of these information to set up a framework for one-dimensional measures to discretize the equations, and use the finite element and central difference methods to obtain numerical approximations of the weak solutions. We also show that the numerical solutions converge to the actual solution and obtain the rate of convergence.
Fractal; Laplacian; wave equation; self-similar measure with overlaps.
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@inproceedings{wei2021strong,
  title={Strong damping wave equations defined by a class of self-similar measures with overlaps,},
  author={Wei Tang, and Zhiyong Wang},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20210816102931489060857},
  year={2021},
}
Wei Tang, and Zhiyong Wang. Strong damping wave equations defined by a class of self-similar measures with overlaps,. 2021. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20210816102931489060857.
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