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Strong instability of solitary-wave solutions of a generalized Boussinesq equation. (English) Zbl 0973.35163
The author examines a generalized Boussinesq equation \[ u_{tt}- u_{xx}+ (u_{xx}+ f(u))_{xx}= 0,\quad x\in\mathbb{R},\quad t>0 \] with \(f\in C^1(\mathbb{R})\). This model arises by describing water waves, phase transitions for shape memory alloys, and in the theory of anharmonic lattice waves. It is known that the above equation possesses travelling-wave type solutions with finite energy, which are called here solitary-wave solutions. First, by constructing a special functional of the solution, the author proves strong instability of solitary-wave solutions. Then some improved blow-up results are established in the particular case \(f(u)=|u|^{p-1} u\), \(p>1\).

35Q51 Soliton equations
35B35 Stability in context of PDEs
37K45 Stability problems for infinite-dimensional Hamiltonian and Lagrangian systems
76B25 Solitary waves for incompressible inviscid fluids
Full Text: DOI
[1] J. C. Alexander, and, R. Sachs, Liner instability of solitary waves of the Boussinesq-type equation: A computer assited computation, preprint.
[2] J. Angulo, Improved blow-up of solutions of the generalized Boussinesq equation (GBQ), preprint.
[3] Bona, J.; Sachs, R., Global existence of smooth solutions and stability of solitary waves for a generalized Boussinesq equation, Comm. math. phys., 118, 15-29, (1988) · Zbl 0654.35018
[4] Boussinesq, J., Théorie des ondes et de remous qui se propagent…, J. math. pures appl. (2), 17, 55-108, (1872) · JFM 04.0493.04
[5] R. E. Caflisch, Shallow water waves, lecture notes, New York University, New York.
[6] Falk, F.; Laedke, E.W.; Spatschek, K.H., Stability of solitary wave pulses in shape-memory alloys, Phys. rev. B, 36, 3031-3041, (1987)
[7] Grillakis, M.; Shatah, J.; Strauss, W., Stability theory of solitary waves in the presence of symmetry, I, J. funct. anal., 74, 160-197, (1987) · Zbl 0656.35122
[8] Kalantarov, V.; Ladyzhenskaya, O., The occurrence of collapse for quasi-linear equations of parabolic and hyperbolic types, J. soviet math., 10, 53-70, (1978) · Zbl 0388.35039
[9] 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., 192, 1-21, (1974) · Zbl 0288.35003
[10] Liu, Y., Instability of solitary waves for generalized Boussinesq equations, J. dynamics differential equations, 5, 537-558, (1993) · Zbl 0784.34048
[11] Liu, Y., Instability and blow-up of solutions to a generalized Boussinesq equation, SIAM J. math. anal., 26, 1527-1546, (1995) · Zbl 0857.35103
[12] Liu, Y., Decay and scattering of small solutions of a generalized Boussinesq equation, J. funct. anal., 147, 51-68, (1997) · Zbl 0884.35129
[13] Liu, Y., Existence and blow up of a nonlinear pochhammer – chree equation, Indiana univ. math. J., 45, 797-816, (1996) · Zbl 0883.35116
[14] Sachs, R.L., On the blow-up of certain solutions of the “good” Boussinesq equation, Applicable anal., 36, 145-152, (1990) · Zbl 0674.35082
[15] Smith, S.J.; Chatterjee, R.J., Phys. lett. A, 125, 129, (1987)
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