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