Acta mathematica scientia, Series B >
FINITE TIME BLOW UP OF THE SOLUTIONS TO BOUSSINESQ EQUATION WITH LINEAR RESTORING FORCE AND ARBITRARY POSITIVE ENERGY
Received date: 2015-02-05
Revised date: 2015-11-18
Online published: 2016-06-25
Supported by
The authors are partially supported by Grant No. DFNI I-02/9 of the Bulgarian Science Fund.
Finite time blow up of the solutions to Boussinesq equation with linear restoring force and combined power nonlinearities is studied. Sufficient conditions on the initial data for nonexistence of global solutions are derived. The results are valid for initial data with arbitrary high positive energy. The proofs are based on the concave method and new sign preserving functionals.
Nikolay KUTEV , Natalia KOLKOVSKA , Milena DIMOVA . FINITE TIME BLOW UP OF THE SOLUTIONS TO BOUSSINESQ EQUATION WITH LINEAR RESTORING FORCE AND ARBITRARY POSITIVE ENERGY[J]. Acta mathematica scientia, Series B, 2016 , 36(3) : 881 -890 . DOI: 10.1016/S0252-9602(16)30047-9
[1] Christov C I, Marinov T T, Marinova R S. Identification of solitary-wave solutions as an inverse problem: Application to shapes with oscillatory tails. Math Comp Simulation, 2009, 80: 56-65
[2] Mishkis A D, Belotserkovskiy P M. On resonance of an infinite beam on uniform elastic foundation. ZAMM-Z Angew Math Mech, 1999, 79: 645-647
[3] Porubov A. Amplification of Nonlinear Strain Waves in Solids. World Scientific, 2003
[4] Samsonov A M. Strain Solitons in Solids and How to Construct Them. Chapman and Hall/CRC, 2001
[5] Maugin G A. Nonlinear Waves in Elastic Crystals. Oxford University Press, 1999
[6] Falk F, Laedke E W, Spatschek K H. Stability of solitary-wave pulses in shape-memory alloys. Phys Rev B, 1987, 36(6): 3031-3041
[7] Tao T, Visan M, Zhang X. The nonlinear Schrödinger equation with combined power-type nonlinearities. Comm Partial Differential Equations, 2007, 32: 1281-1343
[8] Payne L E, Sattinger D H. Saddle points and instability of nonlinear hyperbolic equations. Israel J Math, 1975, 22(3/4): 273-303
[9] Kutev N, Kolkovska N, Dimova M. Global behavior of the solutions to Boussinesq type equation with linear restoring force. AIP CP, 2014, 1629: 172-185
[10] Xu R. Cauchy problem of generalized Boussinesq equation with combined power-type nonlinearities. Math Meth Appl Sci, 2011, 34: 2318-2328
[11] Liu Y, Xu R. Potential well method for Cauchy problem of generalized double dispersion equations. J Math Anal Appl, 2008, 338: 1169-1187
[12] Polat N, Ertas A. Existence and blow up of solution of Cauchy problem for the generalized damped multidimensional Boussinesq equation. Math Anal Appl, 2009, 349: 10-20
[13] Wang S, Chen G. Cauchy problem of the generalized double dispersion equation. Nonlinear Anal, 2006, 64: 159-173
[14] Yang Zhijian, Guo Boling. Cauchy problem for the multi-dimensional Boussinesq type equation. J Math Anal Appl, 2008, 340: 64-80
[15] Straughan B. Further global nonexistence theorems for abstract nonlinear wave equations. Proc Amer Math Soc, 1975, 48: 381-390
[16] Liu Y. Instability and blow up of solutions to a generalized Boussinesq equation. SIAM J Math Anal, 1995, 26: 1527-1546
[17] Levine H A. Instability and nonexistence of global solutions to nonlinear wave equations of the form Putt=Au+F(u). Trans Amer Math Soc, 1974, 192: 1-21
[18] Kalantarov V K, Ladyzhenskaya O A. The occurrence of collapse for quasilinear equations of parabolic and hyperbolic types. J Soviet Math, 1978, 10(1): 53-70
/
| 〈 |
|
〉 |