具广义非线性源的波动方程的高能爆破

Abstract

Abstract:

This paper investigates the finite time blow up of solutions for the initial boundary value problem of a class of some nonlinear wave equations with general nonlinearity at high initial energy level. By employing the classical concavity method, we establish some new sufficient conditions on initial data such that the solution with arbitrarily positive initial energy blows up in finite time.

Key words: Wave equation, High initial energy, Finite time blow up, General nonlinearity

CLC Number:

O175.29

Yanbing Yang,Wei Lian,Shaobin Huang,Runzhang Xu. Finite Time Blow up of Solutions for Nonlinear Wave Equation with General Nonlinearity for Arbitrarily Positive Initial Energy[J].Acta mathematica scientia,Series A, 2018, 38(6): 1239-1244.

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References 13

1	Sattinger D H . On global solution of nonlinear hyperbolic equations. Arch Ration Mech Anal, 1968, 30: 148- 172 doi: 10.1007/BF00250942
2	Levine H A . Some additional remarks on the nonexistence of global solutions to nonlinear wave equations. SIAM J Math Anal, 1974, 5: 138- 146 doi: 10.1137/0505015
3	Levine H A . Instability and nonexistence of global solutions of nonlinear wave equation of the form Pu_tt=Au + F(u). Trans Amer Math Soc, 1974, 192: 1- 21
4	Ball J M . Remarks on blow-up and nonexistence theorems for nonlinear evolution equations. Quart J Math, 1977, 28: 473- 486 doi: 10.1093/qmath/28.4.473
5	Payne L E , Sattinger D H . Saddle points and instability of nonlinear hyperbolic equations. Israel J Math, 1975, 22: 273- 303 doi: 10.1007/BF02761595
6	Liu Y C , Zhao J S . On potential wells and applications to semilinear hyperbolic equations and parabolic equations. Nonlinear Anal, 2006, 64: 2665- 2687 doi: 10.1016/j.na.2005.09.011
7	Xu R Z . Initial boundary value problem for semilinear hyperbolic equations and parabolic equations with critical initial data. Quart Appl Math, 2010, 68: 459- 468 doi: 10.1090/qam/2010-68-03
8	Esquivel-Avila J A . Blow up and asymptotic behavior in a nondissipative nonlinear wave equation. Appl Anal, 2014, 93: 1963- 1978 doi: 10.1080/00036811.2013.859250
9	Gazzola F , Squassina M . Global solutions and finite time blow up for damped semilinear wave equations. Ann Inst H Poincaré Anal Non Linéaire, 2006, 93: 185- 207
10	Chen H , Liu G W . Global existence, uniform decay and exponential growth for a class of semi-linear wave eqution with strong damping. Acta Math Sci, 2013, 33B (1): 41- 58
11	狄华斐, 尚亚东. 一类带有非线性阻尼项和源项的四阶波动方程整体解的存在性与不存在性. 数学物理学报, 2015, 35A (3): 618- 633 doi: 10.3969/j.issn.1003-3998.2015.03.016
	Di H F , Shang Y D . Global existence and nonexistence of solutions for a class of fourth order wave equation with nonlinear damping and source terms. Acta Math Sci, 2015, 35A (3): 618- 633 doi: 10.3969/j.issn.1003-3998.2015.03.016
12	苏晓, 王书彬. 任意正初始能量状态下半线性波动方程解的有限时间爆破. 数学物理学报, 2017, 37A (6): 1085- 1093 doi: 10.3969/j.issn.1003-3998.2017.06.008
	Su X , Wang S B . Finite time blow-up for the damped semilinear wave equations with arbitrary positive initial energy. Acta Math Sci, 2017, 37A (6): 1085- 1093 doi: 10.3969/j.issn.1003-3998.2017.06.008
13	沈继红, 张明有, 杨延冰, 等. 具阻尼的高维广义Boussinesq方程的Cauchy问题的整体适定性. 数学物理学报, 2014, 34A (5): 1173- 1187
	Shen J H , Zhang M Y , Yang Y B , et al. Global well-posedness of Cauchy problem for damped multidimensional generalized Baussinesq equations. Acta Math Sci, 2014, 34A (5): 1173- 1187

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[5]	WANG Fen-Ling, SHI Dong-Yang. High Accurary Analysis of the Bilinear Element for Nonlinear Dispersion-Dissipative Wave Equations [J]. Acta mathematica scientia,Series A, 2014, 34(6): 1599-1610.
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[7]	WANG Yan-Ping. Blow-up of \|Solutions \|of \|an \|Initial \|Boundary \|Value Problem for a Class of Nonlinear Wave Equation [J]. Acta mathematica scientia,Series A, 2009, 29(4): 1093-1103.
[8]	Wu haigen;Yang han\quad ;Liu Juan. Global Existence for Semilinear wave Equations in Exterior Domain [J]. Acta mathematica scientia,Series A, 2007, 27(6): 1089-1097.
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[12]	ZHANG Hong-Wei, CHEN Guo-Wang. Potential Well Method for a Class of Nonlinear Wave Equations of FourthOrder  [J]. Acta mathematica scientia,Series A, 2003, 23(6): 758-768.
[13]	SHANG Ya-Dong, GUO Bo-Ling. The Global Attractors for the Periodic Initial Value Problem for Dissipative Generalized Symmetric Regularized Long Wave Equations [J]. Acta mathematica scientia,Series A, 2003, 23(6): 745-757.
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[15]	Hao Xin-sheng. An Inverse \|problem of \|a Semilinear Wave Equation [J]. Acta mathematica scientia,Series A, 2000, 20(zk): 602-609.

Finite Time Blow up of Solutions for Nonlinear Wave Equation with General Nonlinearity for Arbitrarily Positive Initial Energy

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