一个弱耗散修正的二分量Dullin-Gottwald-Holm系统解的行为研究

Abstract

Abstract:

In this paper, we consider the Cauchy problem of a weakly dissipative modified two-component Dullin-Gottwald-Holm (mDGH2) system. The local well-posedness and the global existence are analyzed, which is to prove that blow-up phenomena cannot happen under the condition $\left(\|y_{0}\|_{L^{2}}^{2}+\|\rho_{0}\|^{2}_{L^{2}}\right)^{\frac{1}{2}}<\frac{4\lambda} {3}.$ We derive the precise blow-up scenario, and then provide several criteria guaranteeing the blow-up of the solutions to the weakly dissipative mDGH2 system. It is worth noting that the solution to the weakly dissipative system is not affected by the weakly dissipative term.

Key words: Two-component peakon system, Weakly dissipative, Blow-up

CLC Number:

O175.2

Shoufu Tian. On the Behavior of the Solution of a Weakly Dissipative Modified Two-Component Dullin-Gottwald-Holm System[J].Acta mathematica scientia,Series A, 2020, 40(5): 1204-1223.

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References 20

1	Camassa R , Holm D . An integrable shallow water equation with peaked solitons. Phys Rev Lett, 1993, 71, 1661- 1664 doi: 10.1103/PhysRevLett.71.1661
2	Camassa R , Holm D , Hyman J . A new integrable shallow water equation. Adv Appl Mech, 1994, 31, 1- 33 doi: 10.1016/S0065-2156(08)70254-0
3	Constantin A , Escher J . Wave breaking for nonlinear nonlocal shallow water equations. Acta Mathematica, 1998, 181, 229- 243 doi: 10.1007/BF02392586
4	Constantin A , Lannes D . The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations. Arch Ration Mech Anal, 2009, 192, 165- 186 doi: 10.1007/s00205-008-0128-2
5	Constantin A , Kolev B . On the geometric approach to the motion of inertial mechanical systems. J Phys A, 2002, 35 (32): R51- R79 doi: 10.1088/0305-4470/35/32/201
6	Fokas A , Fuchssteiner B . Symplectic structures, their Bäcklund transformation and hereditary symmetries. Phys D, 1981, 4, 47- 66 doi: 10.1016/0167-2789(81)90004-X
7	Ghidaglia J M . Weakly damped forced Korteweg-de Vries equations behave as a finite dimensional dynamical system in the long time. J Differential Equations, 1988, 74, 369- 390 doi: 10.1016/0022-0396(88)90010-1
8	Guo F , Gao H , Liu Y . On the wave-breaking phenomena for the two-component Dullin-Gottwald-Holm system. J London Math Soc, 2012, 86, 810- 834 doi: 10.1112/jlms/jds035
9	Guo Z . Blow-up and global solutions to a new integrable model with two components. J Math Anal Appl, 2010, 372, 316- 327 doi: 10.1016/j.jmaa.2010.06.046
10	Holm D , Ó Náraigh L , Tronci C . Singular solutions of a modified two-component Camassa-Holm equation. Phys Rev E, 2009, 79, 016601 doi: 10.1103/PhysRevE.79.016601
11	Ivanov R . Two-component integrable systemsmodelling shallow water waves:The constant vorticity case. Wave Motion, 2009, 46, 389- 396 doi: 10.1016/j.wavemoti.2009.06.012
12	Ionescu-Kruse D . Variational derivation of the Camassa-Holm shallow water equation. J Nonlinear Math Phys, 2007, 14, 303- 312
13	Kato T. Quasi-linear Equations of Evolution, with Applications to Partial Differential Equations//Everitt N. Spectral Theory and Differential Equations. Berlin: Springer-Verlag, 1975
14	Mustafa O G . Existence and uniqueness of low regularity solutions forthe Dullin-Gottwald-Holm equation. Comm Math Phys, 2006, 265, 189- 200 doi: 10.1007/s00220-006-1532-9
15	Novruzov E . Blow-up of solutions for the dissipative Dullin-Gottwald-Holm equation with arbitrary coefficients. J Differential Equations, 2016, 261, 1115- 1127 doi: 10.1016/j.jde.2016.03.034
16	Ott E , Sudan RN . Damping of solitary waves. Phys Fluids, 1970, 13, 1432- 1434 doi: 10.1063/1.1693097
17	Tian L , Gui G , Liu L . On the Cauchy problem and the scattering problem for the Dullin-Gottwald-Holm equation. Comm Math Phys, 2005, 257 (3): 667- 701
18	Zhu M , Xu J . On the wave-breaking phenomena for the periodic two-component Dullin-Gottwald-Holm system. J Math Anal Appl, 2012, 391, 415- 428 doi: 10.1016/j.jmaa.2012.02.058
19	Zhai P P , Guo Z G , Wang W M . Wave breaking phenomenon for a modified two-component Dullin-Gottwald-Holm equation. J Math Phys, 2014, 55, 093101 doi: 10.1063/1.4894368
20	Zhou Y . Blow-up of solutions to the DGH equation. J Funct Anal, 2007, 250, 227- 248 doi: 10.1016/j.jfa.2007.04.019

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On the Behavior of the Solution of a Weakly Dissipative Modified Two-Component Dullin-Gottwald-Holm System

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