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Instability and blow-up of solutions to a generalized Boussinesq equation. (English) Zbl 0857.35103

Summary: We investigate conditions for the finite-time blow-up of solutions of the generalized Boussinesq equation \[ u_{tt}-u_{xx}+ (f(u)+u_{xx})_{xx}=0, \qquad x\in\mathbb{R}, \quad t>0. \] The conditions are expressed in terms of the energy of the ground state. In particular, there exist initial data arbitrarily close to the stationary state of lowest energy whose solutions blow up in finite time.

MSC:

35Q35 PDEs in connection with fluid mechanics
35B40 Asymptotic behavior of solutions to PDEs
76B25 Solitary waves for incompressible inviscid fluids
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