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Global existence and blow up of solutions for Cauchy problem of generalized Boussinesq equation. (English) Zbl 1185.35192

Summary: We study the Cauchy problem for the generalized Boussinesq equation \(u_{tt} - u_{xx}+(u_{xx}+f(u))_{xx}=0\), where \(f(u)=\pm |u|^p\) or \(\pm |u|^{p - 1}u\), \(p>1\). By introducing a family of potential wells we obtain invariant sets, vacuum isolating and threshold result of global existence and nonexistence of solution.

MSC:

35Q35 PDEs in connection with fluid mechanics
35B35 Stability in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
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