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Potential well method for initial boundary value problem of the generalized double dispersion equations

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  • In this paper we study the initial boundary value problem of the generalized double dispersion equations $u_{t t}-u_{x x}-u_{x x t t}+u_{x x x x}=f(u)_{x x}$, where $f(u)$ include convex function as a special case. By introducing a family of potential wells we first prove the invariance of some sets and vacuum isolating of solutions, then we obtain a threshold result of global existence and nonexistence of solutions. Finally we discuss the global existence of solutions for problem with critical initial condition.
    Mathematics Subject Classification: Primary: 35M20, 35Q72.

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