2023年11月第60卷第6期四川大学学报（自然科学版）JournalofSichuanUniversity(NaturalScienceEdition)Nov.2023Vol.60No.6具内部阻尼项的Kirchhoff板方程的长时行为田佳鑫，付晓玉（四川大学数学学院，成都610064）摘要：本文研究了具有局部内部阻尼项的Kirchhoff板方程的长时间行为．在该方程中，边界条件包含0阶和2阶的齐次Dirichlet边界条件或1阶和3阶的Robin边界条件：在关于阻尼的一些基本假设下，本文证明了方程的解具有对数衰减性。关键词：Kirchhoff板方程；Carleman估计；预解式估计中图分类号：0231.4LongtimebehaviorofKirchhoffplateequationswithinternaldamping文献标识码：ADOI:10.19907/j.0490-6756.2023.061005TIANJia-Xin,FUXiao-Yu(SchoolofMathematics,SichuanUniversity,Chengdu610064,China)Abstract:LongtimebehavioroftheKirchhoffplateequationswithlocallydistributedinternaldampingisstudied.TheequationisequippedwitheitherhomogeneousDirichletboundaryconditionsofzeroorderandsecondorder,orRobinboundaryconditionsoffirstorderandthirdorder.Undersomeassumptionsonthedamping,itisshownthatsufficientlysmoothsolutionsoftheequationdecaylogarithmically.Keywords:Kirchhoffplateequation;Carlemanestimate;Resolventestimate(2010MSC93B05)LetbeaboundeddomaininR"(nEN)1IntroductionwithCboundary.Denotebyu=(urn",Ua,)the(1)Inrecentyears,thelogarithmicstabilityofwave/plateequationsisstudiedextensively(seeRefs.[1-5]andthereferencestherein).Forin-stance,thelogarithmicdecayestimatesfortheKirchhoffplateequationsandtheBernoulli-Eulerplateequationswithboundarydampingareinves-tigatedinRef.[1].Ontheotherhand,forthelogarithmicsta-bilityoftheKirchhoffplateequationswithlocallydistributeddamping,thereareveryfewrefer-ences.Inthispaper,westarttodealwiththisproblem.收稿日期：2023-03-04基金项目：国家自然科学基金（11971333）作者简介：田佳鑫（1995一），男，重庆市陵人，博士研究生，主要研究方向为分布参数系统的控制理论。E-mail:tjx20102495@163.comunitoutwardnormalvectorofatα=(αi,*",an)EI.Considerthefollowingdampedplatee-quation[zu-zu+z+dz=0,inR+X,((z(0),z(0))=(α,之l),in2where>0andthedampingfunctiond(·)EL（;R+).Additionally,equation(1)isimposedth...
