GLOBAL EXISTENCE AND POINTWISE ESTIMATES OF SOLUTIONS TO GENERALIZED KURAMOTO-SIVASHINSKY SYSTEM IN MULTI-DIMENSIONS

doi:10.1007/s10473-019-0115-8

数学物理学报(英文版) ›› 2019, Vol. 39 ›› Issue (1): 180-194.doi: 10.1007/s10473-019-0115-8

GLOBAL EXISTENCE AND POINTWISE ESTIMATES OF SOLUTIONS TO GENERALIZED KURAMOTO-SIVASHINSKY SYSTEM IN MULTI-DIMENSIONS

张颖姝, 李朗, 房少梅

Department of Mathematics, South China Agricultural University, Guangzhou 510642, China

收稿日期:2017-09-25 出版日期:2019-02-25 发布日期:2019-03-13
通讯作者: Shaomei FANG E-mail:dz90@scau.edu.cn
作者简介:Yingshu ZHANG,yinshu728@sina.com;Lang LI, lilang@scau.edu.cn
基金资助:
This work was supported by the National Natural Science Foundation of China (11271141), and Chongqing Science & Technology Commission (cstc2018jcyjAX0787).

GLOBAL EXISTENCE AND POINTWISE ESTIMATES OF SOLUTIONS TO GENERALIZED KURAMOTO-SIVASHINSKY SYSTEM IN MULTI-DIMENSIONS

Yingshu ZHANG, Lang LI, Shaomei FANG

Department of Mathematics, South China Agricultural University, Guangzhou 510642, China

Received:2017-09-25 Online:2019-02-25 Published:2019-03-13
Contact: Shaomei FANG E-mail:dz90@scau.edu.cn
Supported by:
This work was supported by the National Natural Science Foundation of China (11271141), and Chongqing Science & Technology Commission (cstc2018jcyjAX0787).

摘要/Abstract

摘要： The Cauchy problem of the generalized Kuramoto-Sivashinsky equation in multi-dimensions (n ≥ 3) is considered. Based on Green's function method, some ingenious energy estimates are given. Then the global existence and pointwise convergence rates of the classical solutions are established. Furthermore, the L^p convergence rate of the solution is obtained.

关键词: Kuramoto-Sivashinsky equation, global existence, pointwise estimates, Green's function method, energy method

Abstract: The Cauchy problem of the generalized Kuramoto-Sivashinsky equation in multi-dimensions (n ≥ 3) is considered. Based on Green's function method, some ingenious energy estimates are given. Then the global existence and pointwise convergence rates of the classical solutions are established. Furthermore, the L^p convergence rate of the solution is obtained.

Key words: Kuramoto-Sivashinsky equation, global existence, pointwise estimates, Green's function method, energy method

张颖姝, 李朗, 房少梅. GLOBAL EXISTENCE AND POINTWISE ESTIMATES OF SOLUTIONS TO GENERALIZED KURAMOTO-SIVASHINSKY SYSTEM IN MULTI-DIMENSIONS[J]. 数学物理学报(英文版), 2019, 39(1): 180-194.

Yingshu ZHANG, Lang LI, Shaomei FANG. GLOBAL EXISTENCE AND POINTWISE ESTIMATES OF SOLUTIONS TO GENERALIZED KURAMOTO-SIVASHINSKY SYSTEM IN MULTI-DIMENSIONS[J]. Acta mathematica scientia,Series B, 2019, 39(1): 180-194.

参考文献

[1] Kuramoto Y. Instability and turbulence of wave fronts in reaction-diffusion systems. Progr Theoret Phys, 1980, 63(6):1885-1903
[2] Sivashinsky G I. Nonlinear analysis of hydrodynamic instability in laminar flames-I Derivation of basic equations. Acta Astronaut, 1977, 4(11/12):1177-1206
[3] LaQuey R E, Mahajan S M, Rutherford P H, et al. Nonlinear saturation of the trapped-ion mode. Phys Rev Lett, 1975, 34(7):391-394
[4] Sivashinsky G I, Michelson D M. On irregular wavy flow of a liquid film down a vertical plane. Progr Theoret Phys, 1980, 63(6):2112-2114
[5] Lin J, Kahn P B. Phase and amplitude instability in delay-diffusion population models. J Math Biol, 1982, 13(3):383-393
[6] Shlang T, Sivashinsky G I. Irregular flow of a liquid film down a vertical column. J Phys (Paris), 1982, 43(3):459-466
[7] Cerpa E, Mercado A. Local exact controllability to the trajectories of the 1-D Kuramoto-Sivashinsky equation. J Differential Equations, 2011, 250(4):2024-2044
[8] Cousin A T, Larkin N A. Kuramoto-Sivashinsky equation in domains with moving boundaries. Port Math (NS), 2002, 59(3):335-349
[9] Kaikina E I. Subcritical Kuramoto-Sivashinsk-type equation on a half-line. J Differential Equations, 2006, 220(2):279-321
[10] Nicolaenko B, Scheurer B. Remarks on the Kuramoto-Sivashinsky equation. Phys D, 1984, 12(1/3):391-395
[11] Giacomelli L, Otto F. New bounds for the Kuramoto-Sivashinsky equation. Commun Pure Appl Anal, 2010, 58(3):297-318
[12] Armaou A, Christofides P D. Feedback control of the Kuramoto-Sivashinsky equation. Phys D, 2000, 137(1/2):49-61
[13] Cerpa E. Null controllability and stabilization of the linear Kuramoto-Sivashinsky equation. Commun Pure Appl Anal, 2010, 9(1):91-102
[14] Guo B L. The existence and nonexistence of a global smooth solution for the initial value problem of generalized Kuramoto-Sivashinsky type equations. J Math Res Exposition, 1991, 11(1):57-70
[15] Zhang L H. Decay of solutions of the multidimensional generalized kuramoto-sivashinsky system. IMA J Appl Math, 1993, 50(1):29-42
[16] Allouba H. From Brownian-time Brownian sheet to a fourth order and a Kuramoto-Sivashinsky-variant interacting PDEs systems. Stoch Anal Appl, 2011, 29(6):933-950
[17] Zhao H J, Tang S Q. Nonlinear stability and optimal decay rate for a multidimensional generalized Kuramoto-Sivashinsky system. J Math Anal Appl, 2000, 246(2):423-445
[18] Shi R K, Wang W K. The pointwise estimates of solutions to the Cauchy problem of a chemotaxis model. Chinese Ann Math Ser B, 2016, 37(1):111-124
[19] Chen J, Li Y C, Wang W K. Global classical solutions to the Cauchy problem of conservation laws with degenerate diffusion. J Differential Equations, 2016, 260(5):4657-4682
[20] Wang W K. Nonlinear evolution systems and green's function. Acta Math Sci, 2010, 30B(6):2051-2063
[21] Li N Y, Mi L F. Pointwise estimates of solutions for the cahn-hilliard equation with inertial term in multidimensions. J Math Anal Appl, 2013, 397(1):75-87
[22] Liu M, Wang W K. Global existence and pointwise estimates of solutions for the multidimensional generalized Boussinesq-type equation. Commun Pure Appl Anal, 2014, 13(3):1203-1222

GLOBAL EXISTENCE AND POINTWISE ESTIMATES OF SOLUTIONS TO GENERALIZED KURAMOTO-SIVASHINSKY SYSTEM IN MULTI-DIMENSIONS

GLOBAL EXISTENCE AND POINTWISE ESTIMATES OF SOLUTIONS TO GENERALIZED KURAMOTO-SIVASHINSKY SYSTEM IN MULTI-DIMENSIONS

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