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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 \(f(u)=|u|^{p-1} u\), \(p>1\).

MSC:
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
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