具有弱阻尼项的广义IMBq方程初值问题解的整体存在性

Abstract

Abstract:

In this paper, we prove that the Cauchy problem for the generalized IMBq equations with weak damping, $\100dpi \inline $u_{tt}-u_{xx}-u_{xxtt}+\nu_0u_t=f(u)_{xx}$$ , $\100dpi \inline $x\in {\Bbb R}$,$t>0$$ , admits a unique global generalized solution in $\100dpi \inline $C^2([0,\infty);H^s({{\Bbb R}}))(s\geq 2$$ ~is~real~number) and a unique global classical solution in $\100dpi \inline $C^2([0,\infty);$$H^s({{\Bbb R}}))(s>\frac{7}{2})$$ . We give the sufficient conditionsof the blow up of the solution for the above mentioned Cauchy problem.

Key words: A generalized IMBq equation with weak damping, Cauchy problem , Global solution , Blow up of solution

CLC Number:

35G25

Cite this article

CHEN Xiang-Ying. Global Existence of Solution of Cauchy Problem for A Generalized IMBq Equation with Weak Damping[J].Acta mathematica scientia,Series A, 2015, 35(1): 68-82.

share this article

0
/ / Recommend

Add to citation manager EndNote|Reference Manager|ProCite|BibTeX|RefWorks

URL: http://121.43.60.238/sxwlxbA/EN/

http://121.43.60.238/sxwlxbA/EN/Y2015/V35/I1/68

Trendmd

References

[1] Ar\'{e}valo E, Gaididei Y, Mertens F G. Soliton dynamics in damped and forced Boussinesq equations. Eur Phys J B,2002, 27: 63--74
[2] Makhankov V G. Dynamics of classical solitons (in non-integrable systems). Phys Rep Lett C, 1978, 35(1): 1--128
[3]}Chen G, Wang S. Proceedings of the Conference on Nonlinear Partial Differential Equations and Applications. Singapore: World Scientific, 1998

[4] Constantin A, Molinet L. The initial value problem for a generalized Boussinesq equation. Diff Int Eqns, 2002, 15(9): 1061--1072

[5] Clarkson A, LeVeque R J, Saxton R.Solitary-wave interactions in elastic rods. Stud Appl Math, 1986, 75: 95--122

[6] Liu Y. Existence and blow-up of solution of a nonlinear Pochhammer-Chree equation. Indiana Univ Math J, 1996, 45: 797--816

[7] Chen G, Rui W, Chen X.Cauchy problem for a damped generalized IMBq equation. J Math Phys, 2011, 52: 053504-19

[8] Wang S, Chen G. Small amplitude solutions of the generalized IMBq equation. J Math Anal and Appl, 2002, 274: 846--866

[9] Kato T, Pouce G. Commutator estimates and the Euler and Navier-Stokes equations. Commun Pure Appl Math, 1988, 41: 891--907

[10] Hrusa W J. Global existence and asymptotic stability for a similinear hyperbolic Volterra equation with large initial data. SIAM J Math Anal, 1985, 16(1): 110--134

[11] 陈国旺. 索伯列夫空间导论. 北京:科学出版社, 2013

[12] Kalantarov V K, Ladyzhenskaya O A. The occurrence of collapse for quasilinear equation of parabolic and hyperbolic types. J Sov Math, 1978, 10: 53--70

Related Articles 15

[1]	Wang Jun, Wang Tianlu, Wen Yanhua, Zhou Xianfeng. Global Solutions of IVP for N-Dimensional Nonlinear Fractional Differential Equations with Delay [J]. Acta mathematica scientia,Series A, 2018, 38(5): 903-910.
[2]	Wang Li. The Inverse Problem for the Supersonic Plane Flow Past a Curved Wedge [J]. Acta mathematica scientia,Series A, 2018, 38(4): 679-686.
[3]	Ye Yaojun, Hu Yue. Global Existence and Exponential Decay of Solution for Some Higher-Order Wave Equation [J]. Acta mathematica scientia,Series A, 2018, 38(1): 110-121.
[4]	Xie Ou, Meng Zehong, Zhao Zhenyu, You Lei. A Truncation Method Based on Hermite Functions Expansion for a Cauchy Problem of the Laplace Equation [J]. Acta mathematica scientia,Series A, 2017, 37(3): 457-468.
[5]	Zhang Hongwei, Liu Gongwei, Hu Qingying. Initial Boundary Value Problem for a Class Wave Equation with Logarithmic Source Term [J]. Acta mathematica scientia,Series A, 2016, 36(6): 1124-1136.
[6]	Deng Jiqin, Deng Ziming. Existence and Uniqueness of Solutions for Nonlocal Cauchy Problem for Fractional Evolution Equations [J]. Acta mathematica scientia,Series A, 2016, 36(6): 1157-1164.
[7]	Lei Qian, Li Ning, Yang Han. Blow Up and Asymptotic Behavior in the Beam Equation [J]. Acta mathematica scientia,Series A, 2015, 35(4): 738-747.
[8]	Ling Zhengqiu, Qin Siqian. Coupled Semilinear Parabolic System with Nonlinear Nonlocal Boundary Condition [J]. Acta mathematica scientia,Series A, 2015, 35(2): 234-244.
[9]	SHEN Ji-Hong, ZHANG Ming-You, YANG Yan-Bing, LIU Bo-Wei, XU Run-Zhang. Global Well-posedness of Cauchy Problem for Damped Multidimensional Generalized Boussinesq Equations [J]. Acta mathematica scientia,Series A, 2014, 34(5): 1173-1187.
[10]	ZHU Xu-Sheng, LI Fang-E. The Global Solutions of the Compressible Euler Equations With Nonlinear Damping in 3-Dimensions [J]. Acta mathematica scientia,Series A, 2014, 34(5): 1111-1122.
[11]	SHU Xiao-Bao, DAI Bin-Xiang. S-asymptotically ω-periodic Solutions of Semi-linear Neutral Fractional Differential Equations [J]. Acta mathematica scientia,Series A, 2014, 34(1): 16-26.
[12]	XU Hong-Mei, XU Ying-Ying. Pointwise Estimates of Solutions of Generalized Benjamin-Bona-Mahony Equations in Multi Dimensions [J]. Acta mathematica scientia,Series A, 2013, 33(6): 1112-1121.
[13]	CHEN Shu-Hong, GUO Bo-Ling. The Existence of Global Solution of the Ginzburg-Landau Equations for Atomic Fermi Gases Near the BCS-BEC Crossover [J]. Acta mathematica scientia,Series A, 2011, 31(5): 1359-1368.
[14]	SHI Qi-Hong, WANG Shu, WANG Chang-Wei. On the Existence of Weak Solutions for Nonlinear Schr\"{o}dinger-Klein-Gordon Equations [J]. Acta mathematica scientia,Series A, 2011, 31(4): 1122-1132.
[15]	YANG Zhi-Jian, CHENG Jian-Ling. Asymptotic Behavior of Solutions to the Kirchhoff Type Equation [J]. Acta mathematica scientia,Series A, 2011, 31(4): 1008-1021.

Metrics

Viewed

Full text

Abstract

Cited

Shared

Discussed

Comments

Recommended 10

[1]	Wang Jinping. Some Results on the Exponential Radon Transform[J]. Acta mathematica scientia,Series A, 2006, 26(1): 31 -038 .
[2]	Chen Shouquan; Lin Zhengyan. Almost Sure Central Limit Theorems for Functions of Random Variables[J]. Acta mathematica scientia,Series A, 2008, 28(4): 747 -756 .
[3]	Yin Chuancun; Zhao Xianghua; Hu Feng. Ladder Height and Supremum of a Random Walk with Applications in Risk Theory[J]. Acta mathematica scientia,Series A, 2009, 29(1): 38 -47 .
[4]	Liu Xiaoji. On the Weighted Moore-Penrose Inverse of the Matrices[J]. Acta mathematica scientia,Series A, 2006, 26(6): 993 .
[5]	Liu Xiaopeng; Liu Kunhui. The Changes of F Distribution Density Curve[J]. Acta mathematica scientia,Series A, 2007, 27(2): 331 -342 .
[6]	Xu Yumei. BREAKDOWN OF CLASSICAL SOLUTION TO A KIND OF QUASILINEAR NON-STRICTLY HYPERBOLIC SYSTEMS[J]. Acta mathematica scientia,Series A, 2006, 26(6): 1089 .
[7]	Huang Yuanqiu; Zhao Tinglei. ON THE CROSSING NUMBER OF THE COMPLETE TRIPARTITE GRAPH K_1,8,n[J]. Acta mathematica scientia,Series A, 2006, 26(6): 1115 .
[8]	Yang Guijun; Liu Minqian; Zhang Runchu. A NOTE ON 2_IV^m-p DESIGNS WITH THE MAXIMUM NUMBER OF CLEAR TWO-FACTOR INTERACTIONS[J]. Acta mathematica scientia,Series A, 2006, 26(6): 1153 .
[9]	Han Junqiang; Niu Pengcheng. Several Hardy Type Inequalities of Generalized Greiner Operator[J]. Acta mathematica scientia,Series A, 2007, 27(1): 57 -066 .
[10]	Luo Luo. Multipliers Between Hardy Spaces on the Unit Ball[J]. Acta mathematica scientia,Series A, 2007, 27(2): 204 -207 .

Abstract
References
Related Articles
Metrics
Comments
Recommended

TOP

Global Existence of Solution of Cauchy Problem for A Generalized IMBq Equation with Weak Damping

PDF (PC)