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.


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