{"title":"Longtime behavior of homoenergetic solutions in the collision dominated regime for hard potentials","authors":"Bernhard Kepka","doi":"10.2140/paa.2024.6.415","DOIUrl":"https://doi.org/10.2140/paa.2024.6.415","url":null,"abstract":"","PeriodicalId":507128,"journal":{"name":"Pure and Applied Analysis","volume":"7 3","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140968844","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We investigate the stability and stabilization of the cubic focusing Klein-Gordon equation around static solutions on the closed ball of radius L in $mathbb{R}^3$. First we show that the system is linearly unstable near the static solution $uequiv 1$ for any dissipative boundary condition $u_t+ au_{nu}=0, ain (0, 1)$. Then by means of boundary controls (both open-loop and closed-loop) we stabilize the system around this equilibrium exponentially under the condition $sqrt{2}Lneq tan sqrt{2}L$. Furthermore, we show that the equilibrium can be stabilized with any rate less than $ frac{sqrt{2}}{2L} log{frac{1+a}{1-a}}$, provided $(a,L)$ does not belong to a certain zero set. This rate is sharp.
我们研究了立方聚焦克莱因-戈登方程在 $mathbb{R}^3$ 中半径为 L 的闭球上静态解附近的稳定性和稳定性。首先，我们证明在任何耗散边界条件 $u_t+ au_{nu}=0, ain (0, 1)$ 下，系统在静态解 $uequiv 1$ 附近是线性不稳定的。然后，通过边界控制（开环和闭环），我们可以在 $sqrt{2}Lneq tan sqrt{2}L$ 的条件下，使系统指数式地稳定在这个平衡点附近。此外，我们还证明，该平衡可以以小于 $fracsqrt{2}}{2L} 的速率稳定下来。log{frac{1+a}{1-a}}$，条件是 $(a,L)$ 不属于某个零集。这个比率非常尖锐。
{"title":"Boundary stabilization of the focusing NLKG equation near unstable equilibria: radial case","authors":"J. Krieger, Shengquan Xiang","doi":"10.2140/paa.2023.5.833","DOIUrl":"https://doi.org/10.2140/paa.2023.5.833","url":null,"abstract":"We investigate the stability and stabilization of the cubic focusing Klein-Gordon equation around static solutions on the closed ball of radius L in $mathbb{R}^3$. First we show that the system is linearly unstable near the static solution $uequiv 1$ for any dissipative boundary condition $u_t+ au_{nu}=0, ain (0, 1)$. Then by means of boundary controls (both open-loop and closed-loop) we stabilize the system around this equilibrium exponentially under the condition $sqrt{2}Lneq tan sqrt{2}L$. Furthermore, we show that the equilibrium can be stabilized with any rate less than $ frac{sqrt{2}}{2L} log{frac{1+a}{1-a}}$, provided $(a,L)$ does not belong to a certain zero set. This rate is sharp.","PeriodicalId":507128,"journal":{"name":"Pure and Applied Analysis","volume":"21 17","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141211688","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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