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Blow-up of solutions of a class of generalized Boussinesq equations. (Chinese) Zbl 0960.35090

Summary: Applying the Fourier transform method, we study the existence of local solutions and the nonexistence of global solutions to initial-boundary value problems for a class of generalized Boussinesq equations: \[ u_{tt}- u_{xx}= bu_{xxxx}+ a(u^p)_{xx}+ cu^q, \] where \(b> 0\), \(a\) and \(c\) are arbitrary real numbers, and \(p\leq 1\), \(q\geq 1\) are integers. We obtain the existence of local solutions to these problems and some sufficient conditions for the blow-up of the solutions within a finite time, and give a few concrete examples.

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35B40 Asymptotic behavior of solutions to PDEs
35A07 Local existence and uniqueness theorems (PDE) (MSC2000)
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