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Potential well method for Cauchy problem of generalized double dispersion equations. (English) Zbl 1140.35011

Summary: We study the Cauchy problem of generalized double dispersion equations \(u_{tt}- u_{xx}- u_{xxtt}+ u_{xxxx}= f(u)_{xx}\), where \(f(u)=|u|^p\), \(p> 1\) or \(u^{2k}\), \(k= 1,2,\dots\). By introducing a family of potential wells we not only get a threshold result of global existence and nonexistence of solutions, but also obtain the invariance of some sets and vacuum isolating of solutions. In addition, the global existence and finite time blow up of solutions for problem with critical initial conditions \(E(0)= d\), \(I(u_0)\geq 0\) or \(I(u_0)< 0\) are proved.

MSC:

35L75 Higher-order nonlinear hyperbolic equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35L30 Initial value problems for higher-order hyperbolic equations
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