强阻尼波动方程吸引子的正则性及其逼近

Abstract

Abstract:

In this paper the authors consider the initial boundary value problem of the strongly damped wave equation. By using the analytic property of the semigroup generated by the principal operator of the equation in the phase space, they show the smoothing effect of the solution, which is much different from the case of weakly dissipative wave equations. Thus they obtain the regularity of the global attractor, and construct inertial manifolds of the equation, just as usually done in the selfadjoint cases.

Key words: Ｓｔｒｏｎｇｌｙｄａｍｐｅｄｗａｖｅｅｑｕａｔｉｏｎ, Ａｎａｌｙｔｉｃｓｅｍｉｇｒｏｕｐ, Ｒｅｇｕｌａｒｉｔｙ, Ｇｌｏｂａｌａｔｔｒａｃｔｏｒ, Ａｐｐｒｏｘｉｍａｔｅｉｎｅｒｔｉａｌｍａｎｉｆｏｌｄ

CLC Number:

３５Ｌ２０，３５Ｂ４０，５８Ｆ３９

Li Yongsheng Zhang Weiguo. Regularity and Approximation of the Attractor for the Strongly Damped Wave Equation[J].Acta mathematica scientia,Series A, 2000, 20(3): 342-350.

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References

１　ＢａｂｉｎＡＶ，ＶｉｓｈｉｋＭＩ．Ｒｅｇｕｌａｒａｔｔｒａｃｔｏｒｓｏｆｓｅｍｉｇｒｏｕｐｓａｎｄｅｖｏｌｕｔｉｏｎｅｑｕａｔｉｏｎｓ．ＪＭａｔｈＰｕｒｅｓｅｔＡｐｐｌ，１９８３，６２（４）：４４１－４４９
２　ＢａｂｉｎＡＶ，ＶｉｓｈｉｋＭＩ．Ａｔｔｒａｃｔｏｒｓｏｆｅｖｏｌｕｔｉｏｎｅｑｕａｔｉｏｎｓ．ＮｏｒｔｈＨｏｌｌａｎｄ：Ａｍｓｔｅｒｄａｍ，１９９２
３　ＢｅｒｋａｌｉｅｖＺＢ．Ａｎａｔｔｒａｃｔｏｒｏｆｓｏｍｅｑｕａｓｉｌｉｎｅａｒｓｙｓｔｅｍｏｆｄｉｆｆｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎｓｗｉｔｈｖｉｓｃｏｅｌａｓｔｉｃｔｅｒｍｓ．ＲｕｓｓｉａｎＭａｔｈＳｕｒｖｅｙ，４０（１）：２０９－２１０
４　ＢｅｒｋａｌｉｅｖＺＢ．Ａｎａｔｔｒａｃｔｏｒｏｆａｎｏｎｌｉｎｅａｒｅｖｏｌｕｔｉｏｎｅｑｕａｔｉｏｎｏｆｖｉｓｃｏｅｌａｓｔｉｃｉｔｙ．ＭｏｓｃｏｗＵｎｉｖＭａｔｈＢｕｌｌ，１９８５，４０（５）：６１－６３
５　ＣｈａｎｇＱＳ，ＧｕｏＢＬ．ＦｉｎｉｔｅｄｉｆｆｅｒｅｎｃｅｓｏｌｕｔｉｏｎｆｏｒｎｏｎｌｉｎｅａｒｗａｖｅｅｑｕａｔｉｏｎU_U－U_XX＝（σ（U_X））_X＋U_xxt－f（u）．ＪＣｏｍｐｔＭａｔｈ，１９８４，１２（４）：２９７－３０４
６　ＣｈｏｗＳＮ，ＬｕＫ．ＩｎｖａｒｉａｎｔｍａｎｉｆｏｌｄｓｆｏｒｆｌｏｗｓｉｎＢａｎａｃｈｓｐａｃｅｓ．ＪＤｉｆｆＥｑｓ，１９８８，７４：２８５－３１７
７　ＣｈｕｅｓｈｏｖＩＤ．Ｏｎａｃｏｎｓｔｒｕｃｔｉｏｎｏｆａｐｐｒｏｘｉｍａｔｅｉｎｅｒｔｉａｌｍａｎｉｆｏｌｄｓｆｏｒｓｅｃｏｎｄｏｒｄｅｒｉｎｔｉｍｅｅｖｏｌｕｔｉｏｎｅｑｕａｔｉｏｎｓ．ＮｏｎｌｉｎｅａｒＡｎａｌ，１９９６，２６（５）：１００７－１０２１
８　ＤｅｂｕｓｓｃｈｅＡ，ＭａｒｉｏｎＭ．Ｏｎｔｈｅｃｏｎｓｔｒｕｃｔｉｏｎｏｆｆａｍｉｌｉｅｓｏｆａｐｐｒｏｘｉｍａｔｅｉｎｅｒｔｉａｌｍａｎｉｆｏｌｄｓ．ＪＤｉｆｆＥｑｓ，１９９２，１００：１７３－２０１
９　ＦｉｔｚｇｉｂｂｏｎＷＥ．Ｓｔｒｏｎｇｌｙｄａｍｐｅｄｑｕａｓｉｌｉｎｅａｒｅｖｏｌｕｔｉｏｎｅｑｕａｔｉｏｎｓ．ＪＭａｔｈＡｎａｌＡｐｐｌ，１９８１，７９：５３６－５５０
１０　ＧｈｉｄａｇｌｉａＪＭ，ＴｅｍａｍＲ．Ａｔｔｒａｃｔｏｒｓｆｏｒｄａｍｐｅｄｎｏｎｌｉｎｅａｒｈｙｐｅｒｂｏｌｉｃｅｑｕａｔｉｏｎｓ．ＪＭａｔｈＰｕｒｅｓｅｔＡｐｐｌ，１９８７，６６：２７３－３１９
１１　ＧｈｉｄａｇｌｉａＪＭ，ＴｅｍａｍＲ．ＤｉｍｅｎｓｉｏｎｏｆｔｈｅｕｎｉｖｅｒｓａｌａｔｔｒａｃｔｏｒｄｅｓｃｒｉｂｉｎｇｔｈｅｐｅｒｉｏｄｉｃａｌｌｙｄｒｉｖｅｎｓｉｎｅＧｏｒｄｏｎｅｑｕａｔｉｏｎｓ，ＴｒａｎｓｐｏｒｔＴｈｅｏｒｙＳｔａｔｉｓｔ．Ｐｈｙｓ，１９８７，１６（２～３）：２５３－２６５
１２　ＣｈｅｎＦＸ，ＧｕｏＢＬ，ＺｈａｎｇＰ．Ｌｏｎｇｔｉｍｅｂｅｈａｖｏｉｒｏｆｓｔｒｏｎｇｌｙｄａｍｐｅｄｗａｖｅｅｑｕａｔｉｏｎｓ．ＪＤｉｆｆＥｑｕａｔｉｏｎｓ，１９９８，１４７：２３１－２４１
１３　ＬｏｍｄａｈｌＰＳ，ＳｏｅｒｅｎｓｅｎＯＨ，ＣｈｒｉｓｔｉａｎｓｅｎＰＬ．ＳｏｌｉｔｏｎｅｘｃｉｔａｔｉｏｎｓｉｎＪｏｓｅｐｈｓｏｎｔｕｎｎｅｌｊｕｎｃｔｉｏｎｓ．ＰｈｙｓＲｅｖＢ，１９８２，２５：５７３７－５７４８
１４　ＭａｓｓａｔＰ．Ｌｉｍｉｔｉｎｇｂｅｈａｖｉｏｒｆｏｒｓｔｒｏｎｇｌｙｄａｍｐｅｄｎｏｎｌｉｎｅａｒｗａｖｅｅｑａｔｉｏｎｓ．ＪＤｉｆｆＥｑｓ，１９８３，４８：３３４－３４９
１５　ＮａｇｕｍｏＪ，ＡｒｉｍｏｔｏＪ，ＹｏｓｈｉｚａｗａＳ．Ａｎａｃｔｉｖｅｐｕｌｓｅｔｒａｎｓｍｉｓｓｉｏｎｌｉｎｅｓｉｍｕｌａｔｉｎｇｎｅｒｖｅａｘｏｎ．ＰｒｏｃＩＲＥ，１９６２，５０：２０６１－２０７０
１６　ＰａｏＣＶ．Ａｍｉｘｅｄｉｎｉｔｉａｌｂｏｕｎｄａｒｙｖａｌｕｅｐｒｏｂｌｅｍａｒｉｓｉｎｇｉｎｎｅｕｒｏｐｈｙｓｉｌｏｇｙ．ＪＭａｔｈＡｎａｌＡｐｐｌ，１９７５，５２：１０５－１１９
１７　ＰａｚｙＡ．Ｓｅｍｉｇｒｏｕｐｓｏｆｌｉｎｅａｒｏｐｅｒａｔｏｒｓａｎｄａｐｐｌｉｃａｔｉｏｎｓｔｏｐａｒｔｉａｌｄｉｆｆｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎｓ．Ｂｅｒｌｉｎ：Ｓｐｒｉｎｇｅｒ，１９８３
１８　ＳｅｌｌＧＲ，ＹｏｕＹ．Ｉｎｅｒｔｉａｌｍａｎｉｆｏｌｄｓ：Ｔｈｅｎｏｎｓｅｌｆａｄｊｏｉｎｔｃａｓｅ．ＪＤｉｆｆＥｑｓ，１９９２，９６：２０３－２５５
１９　ＳｈｏｗａｌｔｅｒＲＥ．Ｒｅｇｕｌａｒｉｚａｔｉｏｎａｎｄａｐｐｒｏｘｉｍａｔｉｏｎｏｆｓｅｃｏｎｄｏｒｄｅｒｅｖｏｌｕｔｉｏｎｅｑｕａｔｉｏｎｓ．ＳＩＡＭＪＭａｔｈＡｎａｌ，１９７６，７：４６１－４７２
２０　ＴｅｍａｍＲ．Ｉｎｆｉｎｉｔｅｄｉｍｅｎｓｉｏｎａｌｄｙｎａｍｉｃａｌｓｙｓｔｅｍｓｉｎｍｅｃｈａｎｉｃｓａｎｄｐｈｙｓｉｃｓ．Ｂｅｒｌｉｎ：Ｓｐｒｉｎｇｅｒ，１９８８
２１　ＷｅｂｂＧＦ．Ｅｘｉｓｔｅｎｃｅａｎｄａｓｙｍｐｔｏｔｉｃｂｅｈａｖｉｏｒｆｏｒａｓｔｒｏｎｇｌｙｄａｍｐｅｄｎｏｎｌｉｎｅａｒｗａｖｅｅｑｕａｔｉｏｎ．ＣａｎａｄＪＭａｔｈ，１９８０，３２：６３１－６４３
２２　ＺｈａｎｇＷＧ．ＴｈｅＵｎｉｖｅｒｓａｌｃｏｍｐａｃｔａｔｔｒａｃｔｏｒｆｏｒｍｉｘｅｄｅｑｕａｔｉｏｎｓｏｆｎｏｎｌｉｎｅａｒｗａｖｅａｎｄｎｅｒｖｅｃｏｎｄｕｃｔ．ＡｃｔａＭａｔｈＡｐｐｌＳｉｎｉｃａ，１９９８，２１（３）：３３９－３５２

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Regularity and Approximation of the Attractor for the Strongly Damped Wave Equation

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