GLOBAL EXISTENCE AND POINTWISE ESTIMATES OF SOLUTIONS TO GENERALIZED KURAMOTO-SIVASHINSKY SYSTEM IN MULTI-DIMENSIONS
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摘要
The Cauchy problem of the generalized Kuramoto-Sivashinsky equation in multidimensions(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.
The Cauchy problem of the generalized Kuramoto-Sivashinsky equation in multidimensions(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.
The Cauchy problem of the generalized Kuramoto-Sivashinsky equation in multidimensions(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.
引文
[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] LaQ uey 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
[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] LaQ uey 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