广义Kirchhoff和Boussinesq方程解的动力性质
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摘要
本文的研究工作之一是:对具有非线性边界阻尼和记忆源项的Kirchhoff型对偶波系统,得到了其解适定性及能量的一致衰减估计;当t→∞时,指出了阻尼Kirchhoff型方程的振动解呈现指数衰减.
其二,本文研究了粘性Boussinesq方程,建立了该系统初值问题解的存在性和唯一性定理.在第三章中,得到了粘性Boussinesq系统的一些新结果,推广了Y. Thomas和Li. Congming文[13]中的相应结论.
One of the aims of this study is to develop the well-posedness andthe uniform decay rate of energy for coupled wave equations of Kirchhoff typewith nonlinear boundary damping and memory source term. For the dampedKirchhoff equation, the oscillation solution has been found to be decaiedexponentially in time as t→∞.
Secondly, based on the study of viscous Boussinesq equations, a num-ber of theorems for the existence and uniqueness of solutions to initialvalue problem associated with the equations have been developed. Theresults from the analysis of viscous Boussinesq equations investigated in chapter3 has extended the corresponding theorems in Y. Thomas and Li. Congming [13].
其二,本文研究了粘性Boussinesq方程,建立了该系统初值问题解的存在性和唯一性定理.在第三章中,得到了粘性Boussinesq系统的一些新结果,推广了Y. Thomas和Li. Congming文[13]中的相应结论.
One of the aims of this study is to develop the well-posedness andthe uniform decay rate of energy for coupled wave equations of Kirchhoff typewith nonlinear boundary damping and memory source term. For the dampedKirchhoff equation, the oscillation solution has been found to be decaiedexponentially in time as t→∞.
Secondly, based on the study of viscous Boussinesq equations, a num-ber of theorems for the existence and uniqueness of solutions to initialvalue problem associated with the equations have been developed. Theresults from the analysis of viscous Boussinesq equations investigated in chapter3 has extended the corresponding theorems in Y. Thomas and Li. Congming [13].
引文
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[2] M. M. Cavalcanti, V. N. Domingos Cavalcanti, J. S. Prates Filho and J.A. Soriano , Existence and uniform decay of solutions of a degenerate equa-tion with nonlinear boundary damping and memory source term[J], NonlinearAnalysis T. M. A., 1999,38:281～294.
[3] V. Georgiev and G. Todorova, Existence of a solution of the wave equationswith nonlinear damping and source terms[J], J. of Di?erential Equations,1994,109:295～308.
[4] Shengfan Zhou and Linshan Wang, Kernel sections for damped non-autonomous wave equations with critical exponent[J], Discrete and Contin-uous Dynamical Systems - Series A, 2003,9(2):399～412.
[5] Ryo Ikehata, A note on the global solvability of solutions to some nonlinearwave equations with dissipative terms[J], Di?. Inte. Eqs., 1995,8(3):607～616.
[6] J. M. Ball, Global attractors for damped semilinear wave equations[J], Dis-crete and Continuous Dynamical Systems - Series A, 2004,10(1-2):31～52.
[7] Jorga A. Esquivel-Avila, Qualitative analysis of a nonlinear wave quation[J],Discrete and Continuous Dynamical Systems - Series A, 2004,10(3):787～804.
[8] J. L. Lions, Quelques nethode de R′esolution des Probl′eme aux Limites Non-linearaire[M], Paris: Dunod Gauthier-Villars, 1969.
[9] K. Nishihara, Global existence and asymptotic behavior of the solution ofsome quasilinear hyperbolic equation with linear damping[J], Funk. Ekvaac.,1989,32:343～355.
[10] Y. Yamada, On some quasilinear wave equations with dissipative terms[J],Nagoya Math. J., 1982,87:17～39.
[11] Jeong Ja Bae and Mitsuhiro Nakao, Existence problem for the kirchho? typewave equation with a localized weakly nonlinear dissipqtion in ecterior do-mains[J], Discrete and Continuous Dynamical Systems - Series A, 2004,11(2-3): 731～743.
[12] Jong Yeoul Park and Jeong Ja Bae, On coupled wave equation of Kirchho?type with nonlinear boundary damping and memory term[J], Applied Math-ematics and Computation, 2002,129:87～105.
[13] Y. Thomas and Li. Congming, Global well-posedness of the viscousBoussinesq equations[J], Discrete and Continuous Dynamical Systems,2005,12(1):1～12.
[14] P. Constantin and C. Foias, Navier-stokes Equations[M], chicago Lecturesin Mathematics, Chicage/London: 1988.
[15] H. Brezis and S. Wainger, A note on the limiting cases of Sobolev embeddingand convolution inequalities[J], Comm. P.D.E., 1980,5:773～789.
[16] E. Gagaliardo, E. Gagaliardo, Propriet`a di alcune classi di funzioni inpiu`variaboli[J], Ric. di Mat., 1958,7:102～137.
[17] L. Nirenger, On elliptic partial di?erential equation[J], Ann. Sc. Norm. Pisa,1959,14:115～162.
[18] Abdul Majid Wazwaz, Generalized Boussinesq type of equations with com-pactons, solitons and periodic solutions[J], Applied Mathematics and Com-putation, 2005,167(2):1162～1178.
[19] Hilmi Demiray, The modified reductive perturbation method as applied toBoussinesq equation: strongly dispersive case[J], Applied Mathematics andComputation, 2004,164(1):1～9.
[20] A. Majda and A. Bertozzi, Vorticity and Incompressible Flow[M], Cam-bridge University Press, Cambridge: UK, 2002.
[21] Zhijian Yang and Xia Wang, Blowup of solutions for improved Boussi-nesq type equation[J], Journal of Mathematical Analysis and Applications,2003,278(2):335～353.
[22] Hyung-Chun Lee and O. Yu. Imanuvilov, Analysis of optimal control prob-lems for the 2-D stationary Boussinesq equations[J], Journal of MathematicalAnalysis and Applications, 2000,242(2):191～211.
[23] J. Boussinesq, Comptes Rendus, V. 72, 1871,755～759.
[2] M. M. Cavalcanti, V. N. Domingos Cavalcanti, J. S. Prates Filho and J.A. Soriano , Existence and uniform decay of solutions of a degenerate equa-tion with nonlinear boundary damping and memory source term[J], NonlinearAnalysis T. M. A., 1999,38:281～294.
[3] V. Georgiev and G. Todorova, Existence of a solution of the wave equationswith nonlinear damping and source terms[J], J. of Di?erential Equations,1994,109:295～308.
[4] Shengfan Zhou and Linshan Wang, Kernel sections for damped non-autonomous wave equations with critical exponent[J], Discrete and Contin-uous Dynamical Systems - Series A, 2003,9(2):399～412.
[5] Ryo Ikehata, A note on the global solvability of solutions to some nonlinearwave equations with dissipative terms[J], Di?. Inte. Eqs., 1995,8(3):607～616.
[6] J. M. Ball, Global attractors for damped semilinear wave equations[J], Dis-crete and Continuous Dynamical Systems - Series A, 2004,10(1-2):31～52.
[7] Jorga A. Esquivel-Avila, Qualitative analysis of a nonlinear wave quation[J],Discrete and Continuous Dynamical Systems - Series A, 2004,10(3):787～804.
[8] J. L. Lions, Quelques nethode de R′esolution des Probl′eme aux Limites Non-linearaire[M], Paris: Dunod Gauthier-Villars, 1969.
[9] K. Nishihara, Global existence and asymptotic behavior of the solution ofsome quasilinear hyperbolic equation with linear damping[J], Funk. Ekvaac.,1989,32:343～355.
[10] Y. Yamada, On some quasilinear wave equations with dissipative terms[J],Nagoya Math. J., 1982,87:17～39.
[11] Jeong Ja Bae and Mitsuhiro Nakao, Existence problem for the kirchho? typewave equation with a localized weakly nonlinear dissipqtion in ecterior do-mains[J], Discrete and Continuous Dynamical Systems - Series A, 2004,11(2-3): 731～743.
[12] Jong Yeoul Park and Jeong Ja Bae, On coupled wave equation of Kirchho?type with nonlinear boundary damping and memory term[J], Applied Math-ematics and Computation, 2002,129:87～105.
[13] Y. Thomas and Li. Congming, Global well-posedness of the viscousBoussinesq equations[J], Discrete and Continuous Dynamical Systems,2005,12(1):1～12.
[14] P. Constantin and C. Foias, Navier-stokes Equations[M], chicago Lecturesin Mathematics, Chicage/London: 1988.
[15] H. Brezis and S. Wainger, A note on the limiting cases of Sobolev embeddingand convolution inequalities[J], Comm. P.D.E., 1980,5:773～789.
[16] E. Gagaliardo, E. Gagaliardo, Propriet`a di alcune classi di funzioni inpiu`variaboli[J], Ric. di Mat., 1958,7:102～137.
[17] L. Nirenger, On elliptic partial di?erential equation[J], Ann. Sc. Norm. Pisa,1959,14:115～162.
[18] Abdul Majid Wazwaz, Generalized Boussinesq type of equations with com-pactons, solitons and periodic solutions[J], Applied Mathematics and Com-putation, 2005,167(2):1162～1178.
[19] Hilmi Demiray, The modified reductive perturbation method as applied toBoussinesq equation: strongly dispersive case[J], Applied Mathematics andComputation, 2004,164(1):1～9.
[20] A. Majda and A. Bertozzi, Vorticity and Incompressible Flow[M], Cam-bridge University Press, Cambridge: UK, 2002.
[21] Zhijian Yang and Xia Wang, Blowup of solutions for improved Boussi-nesq type equation[J], Journal of Mathematical Analysis and Applications,2003,278(2):335～353.
[22] Hyung-Chun Lee and O. Yu. Imanuvilov, Analysis of optimal control prob-lems for the 2-D stationary Boussinesq equations[J], Journal of MathematicalAnalysis and Applications, 2000,242(2):191～211.
[23] J. Boussinesq, Comptes Rendus, V. 72, 1871,755～759.