二维空间中半线性波动方程渐近理论的一个新结果

Abstract

Abstract:

The asymptotic theory for a class of semilinear wave equations in two space demensions is established in this paper. The validity of formal approximations on a long time scale of order

C²(J×R²)(J={t|0≤t≤O(|ε|^{-1/(2-k(p-1))}),ε→0,0<2-k(p-1)<1,0

is discussed in the classical sense of C². The time order function obtained in this paper is better than that of [1]～[4].

Key words: Ｓｅｍｉｌｉｎｅａｒｗａｖｅｅｑｕａｔｉｏｎｓ, Ｔｗｏｓｐａｃｅｄｉｍｅｎｓｉｏｎｓ, Ａｓｙｍｐｔｏｔｉｃｔｈｅｏｒｙ, Ａｐｐｌｉｃａｔｉｏｎ．

CLC Number:

６２Ａ１５，６８Ｔ３０

Lai Shaoyong. A New Result of Asymptotic Theory for Semilinear Wave Equations in Two Space Dimensions[J].Acta mathematica scientia,Series A, 2000, 20(2): 261-268.

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１　ＶａｎＨｏｒｓｓｅｎＷＴ．Ａｎａｓｙｍｐｔｏｔｉｃｔｈｅｏｒｙｆｏｒａｃｌａｓｓｏｆｉｎｉｔｉａｌｂｏｕｎｄａｒｙｖａｌｕｅｐｒｏｂｌｅｍｓｆｏｒｗｅａｋｌｙｎｏｎｌｉｎｅａｒｗａｖｅｅｑｕａｔｉｏｎｓｗｉｔｈａｎａｐｐｌｉｃａｔｉｏｎｔｏａｍｏｄｅｌｏｆｔｈｅｇａｌｌｏｐｉｎｇｏｓｃｉｌｌａｔｉｏｎｏｆｏｖｅｒｈｅａｄｔｒａｎｓｍｉｓｓｉｏｎｌｉｎｅｓ．ＳＩＡＭＪＡｐｐｌＭａｔｈ，１９９８，４８（６）：１２２７－１２４３
２　ＶａｎＨｏｒｓｓｅｎＷＴ，ＶａｎＤｅｒＢｕｒｇｈＡＨＰ．Ｏｎｉｎｉｔｉａｌｂｏｕｎｄａｒｙｖａｌｕｅｐｒｏｂｌｅｍｓｆｏｒｗｅａｋｌｙｓｅｍｉｌｉｎｅａｒｔｅｌｅｇｒａｐｈｇｅｑｕａｔｉｏｎ．ａｓｙｍｐｔｏｔｉｃｔｈｅｏｒｙａｎｄａｐｐｌｉｃａｔｉｏｎ，ＳＩＡＭＪＡｐｐｌＭａｔｈ，１９９８，４（４）：７１９－７３７
３　赖绍永．非线性摄动电报方程解的渐近理论及应用．数学物理学报，１９９８，２（１８）：１４９－１５３
４　ＬａｉＳｈａｏｙｏｎｇ，ＭｕＣｈｕｎｌａｉ．Ｔｈｅａｓｙｍｐｔｏｔｉｃｔｈｅｏｒｙｏｆｉｎｉｔｉａｌｖａｌｕｅｐｒｏｂｌｅｍｓｆｏｒｓｅｍｉｌｉｎｅａｒｗａｖｅｅｑｕａｔｉｏｎｓｉｎｔｈｒｅｅｓｐａｃｅｄｉｍｅｎｓｉｏｎｓ．ＡｐｐｌＭａｔｈＪＣＵ，１９９７，１２Ｂ：３２１－３３２
５　莫嘉琪．一类非线性反应扩散方程组的奇摄动．中国科学，Ａ辑，１９８８，１０：１０４１－１０４９
６　莫嘉琪．一类半线性椭圆型方程Ｄｉｒｉｃｈｌｅｔ问题的奇摄动．数学物理学报，１９８７，７：３９５～４０２
７　ＧｕｅｎｔｈｅｒＲＢ，ＪｏｈｎＷＬ．Ｐａｒｉｔａｌｄｉｆｆｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎｓｏｆｍａｔｈｅｍａｔｉｃａｌｐｈｙｓｉｃｓａｎｄｉｎｔｅｇｒａｌｅｑｕａｔｉｏｎｓ．ＰｒｅｎｔｉｃｅＨａｌｌ．１９８８
８　Ｋ＾ｏｊｉＫｕｂｏｔａ．Ｅｘｉｓｔｅｎｃｅｏｆａｇｌｏｂａｌｓｏｌｕｔｉｏｎｔｏａｓｅｍｉｌｉｎｅａｒｗａｖｅｅｑｕａｔｉｏｎｓｗｉｔｈｉｎｉｔｉａｌｄａｔａｏｆｎｏｎｃｏｍｐａｃｔｓｕｐｐｏｒｔｉｎｌｏｗｓｐａｃｅｄｉｍｅｎｓｉｏｎｓ．ＨｏｋｋａｉｄｏＭａｔｈＪｏｕｒｎａｌ，１９９３，２２：１２３－１８０

A New Result of Asymptotic Theory for Semilinear Wave Equations in Two Space Dimensions

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