GLOBAL EXISTENCE AND BLOW-UP OF SOLUTIONS FOR GENERALIZED POCHHAMMER-CHREE EQUATIONS

doi:10.1016/S0252-9602(10)60173-7

数学物理学报(英文版) ›› 2010, Vol. 30 ›› Issue (5): 1793-1807.doi: 10.1016/S0252-9602(10)60173-7

GLOBAL EXISTENCE AND BLOW-UP OF SOLUTIONS FOR GENERALIZED POCHHAMMER-CHREE EQUATIONS

徐润章, 刘亚成

College of Science, Harbin Engineering University, Harbin 150001, China

收稿日期:2006-09-06 出版日期:2010-09-20 发布日期:2010-09-20
基金资助:
This work is supported by National Natural Science Foundation of China (10871055, 10926149); Natural Science Foundation of Heilongjiang
Province (A2007-02; A200810); Science and Technology Foundation of Education Office of Heilongjiang Province (11541276); Foundational
Science Foundation of Harbin Engineering University.

GLOBAL EXISTENCE AND BLOW-UP OF SOLUTIONS FOR GENERALIZED POCHHAMMER-CHREE EQUATIONS

XU Run-Zhang, LIU YA-Cheng

College of Science, Harbin Engineering University, Harbin 150001, China

Received:2006-09-06 Online:2010-09-20 Published:2010-09-20
Supported by:
This work is supported by National Natural Science Foundation of China (10871055, 10926149); Natural Science Foundation of Heilongjiang
Province (A2007-02; A200810); Science and Technology Foundation of Education Office of Heilongjiang Province (11541276); Foundational
Science Foundation of Harbin Engineering University.

摘要/Abstract

摘要：

In this article, we study the initial boundary value problem of generalized Pochhammer-Chree equation
ut_t-u_xx-u_xxt-u_xxtt=f(u)_xx, x ∈Ω, t >0,
u(x, 0)=u₀(x), ut(x, 0)=u₁(x), x ∈Ω,
u(0, t)=u(1, t)=0, t ≥0,
where Ω =(0, 1). First, we obtain the existence of local W_{k, p} solutions. Then, we prove that, if f(s) ∈ in C^k⁺¹(R) is ondecreasing, f(0)=0 and |f(u)| ≤ C₁|u| ∫₀^uf(s)ds + C₂, u₀(x), u₁(x) ∈W^{k, p}(Ω) ∩ W₀{^1, ^p}(Ω), k ≥1, 1< p ≤∞, then for any T>0 the problem admits a unique solution u(x, t) ∈ W^2, ∞ (0, T; W^k, ^p(Ω)∩W₀^1, ^p(Ω) ). Finally, the finite time blow-up of solutions and global W^{k, p} solution of generalized IMBq equations are discussed.

关键词: Pochhammer-Chree equations, initial boundary value problem, W^{k, p} solution, global existence, blow-up

Abstract:

Key words: Pochhammer-Chree equations, initial boundary value problem, W^{k, p} solution, global existence, blow-up

中图分类号:

35L75

徐润章, 刘亚成. GLOBAL EXISTENCE AND BLOW-UP OF SOLUTIONS FOR GENERALIZED POCHHAMMER-CHREE EQUATIONS[J]. 数学物理学报(英文版), 2010, 30(5): 1793-1807.

XU Run-Zhang, LIU YA-Cheng. GLOBAL EXISTENCE AND BLOW-UP OF SOLUTIONS FOR GENERALIZED POCHHAMMER-CHREE EQUATIONS[J]. Acta mathematica scientia,Series B, 2010, 30(5): 1793-1807.

参考文献

[1] Bogolubsky I L. Some examples of inelastic soliton interaction. Comput Phys Commun, 1977, 13(2): 49--55

[2] Clarcson P A, LeVeque R J, Saxton R. Solitary wave interactions in elastic rods. Stud Appl Math, 1986, 75(1): 95--122

[3] Saxton R. Existence of solutions for a finite nonlinearly hyperelastic rod. J Math Anal Appl, 1985, 105(1): 59--75

[4] Li J, Liu Z. Smooth and non-smooth traveling waves in a nonlinearly dispersive equation. Appl Math Model, 2000, 25: 41--56

[5] Zhang W, Ma W. Explicit solitary wave solutions to generalized Pochhammer-Chree equations. Appl Math Mech, 1999, 20(6): 625--632

[6] Li J, Zhang L. Bifurcations of traveling wave solutions in generalized Pochhammer-Chree equation. Chaos, Soliton and Fractals, 2002, 14: 581--593

[7] Rodrigues J F. Obstacle Problems in Mathematical Physics. North-Holland, Amsterdam: Elsevier Science Ltd, 1987

[8] Kocev I.On boundness of L_p derivatives of solutions for elliptic differential equations. Math Sb, 1956, 38: 360--372 (in Russian)

[9] Chen G, Wang S. Existence and nonexistence of global solutions for the generalized IMBq equation. Nonlinear Anal, 1999, 36: 961--980

[10] Naemark M A. Linear Differential Operators. Moscow: Gaustyhisdute, 1954 (in Russian)

[11] Ball J M. Remarks on blow--up and nonexistence theorems for nonlinear evolution equations. Quart J Math, 1977, 28: 473--486

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

[13] Angulo J, Scialom M. Improved blow-up of solutions of a generalized Boussinesq equation. Comput Appl Math, 1999, 18: 333--341, 371

[14] Chae Dongho, Nam Hee-Seok. Local existence and blow-up criterion for the Boussinesq equations. Proc Roy Soc Edinburgh Sect A,
1997, 127: 935--946

[15] Yang Z, Chen G. Blow-up of solutions of a class of generalized Boussinesq equations. Acta Math Sci, 1996, 16(1): 31--39

[16] Straughan B. Global nonexistence of solutions to some Boussinesq type equations. J Math Phys Sci, 1992, 26: 155--164

[17] Sachs R L. On the blow-up of certain solutions of the "good" Boussinesq equation. Appl Anal, 1990, 36: 145--152

[18] Straughan B. Uniqueness and stability for the conduction-diffusion solution to the Boussinesq equations backward in time. Proc Roy Soc
London Ser A, 1975/76, 347: 435--446

[19] Levine H A, Sleeman B D. A note on the nonexistence of global solutions of initial-boundary value problems for the Boussinesq equation u _tt=3u_xxxx+u_xx-12(u²)_xx. J Math Anal Appl, 1985, 107: 206--210

GLOBAL EXISTENCE AND BLOW-UP OF SOLUTIONS FOR GENERALIZED POCHHAMMER-CHREE EQUATIONS

GLOBAL EXISTENCE AND BLOW-UP OF SOLUTIONS FOR GENERALIZED POCHHAMMER-CHREE EQUATIONS

PDF (PC)

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 15

Metrics

本文评价

推荐阅读 10

[1]	Yue-Loong CHANG, Meng-Rong LI, C. Jack YUE, Yong-Shiuan LEE, Tsung-Jui CHIANG-LIN. MATHEMATICAL MODEL FOR THE ENTERPRISE COMPETITIVE ABILITY AND PERFORMANCE THROUGH A PARTICULAR EMDEN-FOWLER EQUATION u"-n^-q-1u (n)^q=0[J]. 数学物理学报(英文版), 2018, 38(4): 1311-1321.
[2]	林勇, 吴艺婷. BLOW-UP PROBLEMS FOR NONLINEAR PARABOLIC EQUATIONS ON LOCALLY FINITE GRAPHS[J]. 数学物理学报(英文版), 2018, 38(3): 843-856.
[3]	孔德兴, 刘琦. GLOBAL EXISTENCE OF CLASSICAL SOLUTIONS TO THE HYPERBOLIC GEOMETRY FLOW WITH TIME-DEPENDENT DISSIPATION[J]. 数学物理学报(英文版), 2018, 38(3): 745-755.
[4]	朱新才. EXISTENCE AND BLOW-UP BEHAVIOR OF CONSTRAINED MINIMIZERS FOR SCHRÖDINGER-POISSON-SLATER SYSTEM[J]. 数学物理学报(英文版), 2018, 38(2): 733-744.
[5]	张晶, 张永前. INITIAL-BOUNDARY PROBLEM FOR THE 1-D EULER-BOLTZMANN EQUATIONS IN RADIATION HYDRODYNAMICS[J]. 数学物理学报(英文版), 2018, 38(1): 34-56.
[6]	刘颖博, Ingo WITT. SMALL DATA SOLUTIONS OF 2-D QUASILINEAR WAVE EQUATIONS UNDER NULL CONDITIONS[J]. 数学物理学报(英文版), 2018, 38(1): 125-150.
[7]	侯飞. ON THE GLOBAL EXISTENCE OF SMOOTH SOLUTIONS TO THE MULTI-DIMENSIONAL COMPRESSIBLE EULER EQUATIONS WITH TIME-DEPENDING DAMPING IN HALF SPACE[J]. 数学物理学报(英文版), 2017, 37(4): 949-964.
[8]	杨凌燕, 李晓光, 吴永洪, Louis CACCETTA. GLOBAL WELL-POSEDNESS AND BLOW-UP FOR THE HARTREE EQUATION[J]. 数学物理学报(英文版), 2017, 37(4): 941-948.
[9]	夏安银, 樊明书, 李珊. BLOW-UP AND LIFE SPAN ESTIMATES FOR A CLASS OF NONLINEAR DEGENERATE PARABOLIC SYSTEM WITH TIME-DEPENDENT COEFFICIENTS[J]. 数学物理学报(英文版), 2017, 37(4): 974-984.
[10]	叶嵎林. GLOBAL CLASSICAL SOLUTION TO THE CAUCHY PROBLEM OF THE 3-D COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH DENSITY-DEPENDENT VISCOSITY[J]. 数学物理学报(英文版), 2016, 36(5): 1419-1432.
[11]	孔慧慧, 李海梁, 张兴伟. A BLOW-UP CRITERION OF SPHERICALLY SYMMETRIC STRONG SOLUTIONS TO 3D COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH FREE BOUNDARY[J]. 数学物理学报(英文版), 2016, 36(4): 1153-1166.
[12]	蒋毅, 罗川. OPTIMAL CONDITIONS OF GLOBAL EXISTENCE AND BLOW-UP FOR A NONLINEAR PARABOLIC EQUATION[J]. 数学物理学报(英文版), 2016, 36(3): 872-880.
[13]	王忠, 崔尚斌. ON THE CAUCHY PROBLEM OF A COHERENTLY COUPLED SCHRÖDINGER SYSTEM[J]. 数学物理学报(英文版), 2016, 36(2): 371-384.
[14]	朱圣国. BLOW-UP OF CLASSICAL SOLUTIONS TO THE COMPRESSIBLE MAGNETOHYDRODYNAMIC EQUATIONS WITH VACUUM[J]. 数学物理学报(英文版), 2016, 36(1): 220-232.
[15]	郑攀, 穆春来, 胡学刚, 张付臣. SECONDARY CRITICAL EXPONENT AND LIFE SPAN FOR A DOUBLY SINGULAR PARABOLIC EQUATION WITH A WEIGHTED SOURCE[J]. 数学物理学报(英文版), 2016, 36(1): 244-256.