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 Lp convergence rate of the solution is obtained.
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
.
DOI: 10.1007/s10473-019-0115-8
[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
