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

Abstract

Abstract: This paper investigates the initial-boundary value problem of the semilinear wave equation with strong damped terms. We provide the sufficient conditions of finite time blow-up of solutions with high initial energy under some reasonable restrictions on the initial data. In the case of the arbitrary positive initial energy, we also present the sufficient conditions of the existence of global solutions in the energy space.

Key words: Damped wave equations, High initial energy, Blow-up, Global existence

CLC Number:

O175

Su Xiao, Wang Shubin. Finite Time Blow-Up for the Damped Semilinear Wave Equations with Arbitrary Positive Initial Energy[J].Acta mathematica scientia,Series A, 2017, 37(6): 1085-1093.

Trendmd

References

[1] Ambroseti A, Rabinowitz P H. Dual variational methods in critical point theory and applications. J Funct Anal, 1973, 14:349-381
[2] Georgiev V, Todorova G. Existence of solutions of the wave equations with nonlinear damping and source terms. J Differ Equa 1994, 109 (2):295-308
[3] Gazzola F, Weth T. Finite time blow-up and global solutions for semilinear parabolic equations with initial data at high energy level. Differ Integral Equ, 2005, 18:961-990
[4] Gazzola F, Squassina M. Global solutions and finite time blow-up for damped semilinear wave equations. Ann I H Poincaré-AN, 2006, 23:185-207
[5] Gerbi S, Houari B S. Exponential decay for solutions to semilinear damped wave equation. Discrete Contin Dyn Syst Ser S, 2012, 5:559-566
[6] Ikehata R, Suzuki T. Stable and unstable sets for evolution equations of parabolic and hyperbolic type. Hiroshima Math J, 1996, 26:475-491
[7] Kutev N, Kolkovska N, Dimova M. Global existence of Caucy problem for Boussinesq paradigm equation. Computer Math Appli, 2013, 65:500-511
[8] Kutev N, Kolkovska N, Dimova M. Finite time blow up of the solutions to Boussinesq equation with linear restoring force and arbitrary positive energy. Acta Math Sci, 2016, 36:881-890
[9] Lions J L, Magenes E. Non-Homogeneous Boundary Value Problems and Applications I. New York:Springer-Verleag, 1972
[10] Liu Y. On potiential wells and vacuum isolating of solution for semilinear wave equations. J Differ Equa, 2003, 192:155-169
[11] Levine H A. Some additional remarks on the nonexistence of global solutions to nonlinear equations. SIAM J Math Anal, 1974, 5:138-146
[12] Levine H A. Instability and nonexistence of global solutions to nonlinear wave equations of the form Pu_tt=-Au+f(u). Trans Amer Math Soc, 1974, 192:1-21
[13] Ohta M. Remarks on blow-up of solutions for nonlinear evolution equations of second order. Adv Math Sci Appl, 19988:901-910
[14] Payne L E, Sattinger D H. Sadle points and instability of nonlinear hyperbolic equations. Israel J Math, 1975, 22:273-303
[15] Sattinger D. On global solution of nonlinear hyperbolic equations. Arch Rat Mech Anal, 1968, 30:148-172
[16] Sun L, Guo B, Gao W. A lower bound for the blow-up time to a damped semilinear wave equation. Appl Math Lett, 2014, 37:22-25
[17] Su X, Wang S. The initial-boundary value problem for the generalized double dispersion equation. Z Angew Math Phys, 2017, 68:53
[18] Vitillaro E. Global nonexistence theorems for a class of evolution equations with dissipation. Arch Ration Mech Anal, 1999, 149 (2):155-182
[19] Wang S, Su X. Global existence and nonexistence of the initial-boundary value problem for the dissipative Boussinesq equation. Nonlinear Anal, 2016, 134:164-188
[20] Xu R, Su J. Global existence and finite time bow-up for a class of semilinear pseudo-parabolic equations. J Funct Anal, 2013, 264:2732-2763
[21] Xu R, Ding Y. Global solutions and finite time blow up for damped Klein-Gordon equation. Acta Math Sci, 2013, 33B(3):643-652
[22] Zhou J. Lower bounds for blow-up time of two nonlinear wave equations. Appl Math Lett, 2015, 45:64-68

Finite Time Blow-Up for the Damped Semilinear Wave Equations with Arbitrary Positive Initial Energy

PDF (PC)

Abstract

Cite this article

share this article

References

Related Articles 15

Metrics

Comments

Recommended 10

[1]	Sun Baoyan. Lower Bound of Blow-Up Time for a p-Laplacian Equation with Nonlocal Source [J]. Acta mathematica scientia,Series A, 2018, 38(5): 911-923.
[2]	Zhao Yuanzhang, Ma Xiangru. Blow-Up Analysis for a Nonlocal Reaction-Diffusion Equation with Variant Diffusion Coefficient [J]. Acta mathematica scientia,Series A, 2018, 38(4): 750-769.
[3]	Liu Dengming, Zeng Xianzhong, Mu Chunlai. A Blow-up Result for a Higher-Order Damped Hyperbolic System with Nonlinear Memory Terms [J]. Acta mathematica scientia,Series A, 2018, 38(2): 304-312.
[4]	Qiu Hua, Zhang Yingshu, Fang Shaomei. The Global Existence Result of a Leray-α-Oldroyd Model to the Incompressible Viscoelastic Flow [J]. Acta mathematica scientia,Series A, 2017, 37(1): 102-112.
[5]	Ma Lingwei, Fang Zhongbo. Lower Bounds of Blow-up Time for a Reaction-Diffusion Equation with Weighted Nonlocal Sources and Robin Type Boundary Conditions [J]. Acta mathematica scientia,Series A, 2017, 37(1): 146-157.
[6]	Yang Lingyan, Li Xiaoguang, Chen Ying. A Sharp Threshold of Blow-Up of a Class of Schrödinger-Hartree Equations [J]. Acta mathematica scientia,Series A, 2016, 36(6): 1117-1123.
[7]	Liu Yang. Potential Well and Application to Non-Newtonian Filtration Equations at Critical Initial Energy Level [J]. Acta mathematica scientia,Series A, 2016, 36(6): 1211-1220.
[8]	Ling Zhengqiu, Wang Zejia. Blow-Up Questions of Solutions to a Class of p-Laplace Equation with Dissipative Gradient Term [J]. Acta mathematica scientia,Series A, 2016, 36(5): 958-964.
[9]	Huang Shoujun, Wang Rui. Plane Wave Solutions to the Euler Equations for Chaplygin Gases [J]. Acta mathematica scientia,Series A, 2016, 36(4): 681-689.
[10]	Bian Dongfen, Tang Tong. Blow-Up of Smooth Solutions to the Compressible MHD Equations [J]. Acta mathematica scientia,Series A, 2016, 36(4): 715-721.
[11]	Di Huafei, Shang Yadong. Global Existence and Nonexistence of Solutions for A Class of Fourth Order Wave Equation with Nonlinear Damping and Source Terms [J]. Acta mathematica scientia,Series A, 2015, 35(3): 618-633.
[12]	Ling Zhengqiu, Qin Siqian. Coupled Semilinear Parabolic System with Nonlinear Nonlocal Boundary Condition [J]. Acta mathematica scientia,Series A, 2015, 35(2): 234-244.
[13]	Wang Hua, He Yijun. Blow-up of Solutions for a Semilinear Hyperbolic Equation with Variable Exponent and Positive Energy [J]. Acta mathematica scientia,Series A, 2015, 35(2): 288-293.
[14]	LING Zheng-Qiu, QIN Sai-Qan. Blow-up and Global Boundedness for a Degenerate Parabolic System with Positive Boundary Value Condition [J]. Acta mathematica scientia,Series A, 2014, 34(5): 1061-1070.
[15]	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.