广义全变差能量相关的伪抛物方程的大时间行为
Large-Time Behavior of Pseudo-Parabolic Equations Associated with Generalized Total Variation Energies
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- Waseda Junior and Senior High School（早稻田大学附属中学及高中）
- Department of Mathematics, Faculty of Education, Chiba University（千叶大学教育学部数学系）
- Division of Mathematics and Informatics, Department of Mathematics and Informatics, Graduate School of Science and Engineering, Chiba University（千叶大学理工学研究科数学信息学专攻）
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中文总结 AI 辅助
本文研究广义全变差能量相关的伪抛物问题，证明了在BV框架下解的存在唯一性，并揭示了其大时间行为收敛于稳态解，尤其在图像处理典型设置中成立。
中文摘要 AI 辅助
本文在$BV$-框架下研究了一个与广义全变差能量相关的伪抛物问题的适定性和大时间行为。一个主要的数学问题源于有限时刻解的Sobolev正则性与相应稳态问题的$BV$-结构之间的不匹配。我们证明了解的存在性和唯一性，并研究了解的大时间行为与稳态问题解之间的关系。此外，我们的结果包括在图像处理中出现的典型设置下，解轨迹收敛到唯一稳态解。
英文摘要
In this paper, we study the well-posedness and large-time behavior of a pseudo-parabolic problem associated with a generalized total variation energy in the $BV$-framework. A main mathematical issue arises from the mismatch between the Sobolev regularity of solutions at finite times and the $BV$-structure of the corresponding steady-state problem. We prove the existence and uniqueness of solutions and investigate the relationship between their large-time behavior and solutions to the steady-state problem. Moreover, our results include the convergence of the solution trajectory to the unique steady-state solution under a typical setting arising in image processing.
