Global existence and nonexistence of solution for Cauchy problem of multidimensional double dispersion equations. (English) Zbl 1176.35119

Summary: We consider the Cauchy problem of multidimensional generalized double dispersion equations \(u_{tt}-\Delta u- \Delta u_{tt}+\Delta^2u= \Delta f(u)\), where \(f(u)=a|u|^p\). By a potential well method we prove the existence and nonexistence of global weak solution without establishing the local existence theory. Furthermore, we derive some sharp conditions for a global existence and the lack of global existence solution.


35L75 Higher-order nonlinear hyperbolic equations
35L82 Pseudohyperbolic equations
35L30 Initial value problems for higher-order hyperbolic equations
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