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Well-posedness, blow-up phenomena, and global solutions for the \(b\)-equation. (English) Zbl 1159.35060
The authors study inital value problems for a multiparameter family of nonlienar dispersive equations (the \(b\)-equation)
\[ u_t - \alpha^2 u_{txx} + c_0 u_x + (b+1)uu_x + \Gamma u_{xxx} = \alpha^2 (b u_x u_{xx} + uu_{xxx}), \] where \(t > 0\), \(x \in {\mathbb R}\), \(u(0,x) = u_0(x)\) is the initial data, and \(c_0, b, \Gamma\), and \(\alpha\) are arbitrary constants. For different values of these constants one obtains such well-known equations as the Korteweg-de Vries equation (\(\alpha = 0\) and \(b=2\)), the Camassa-Holm equation (\(b=2\) and \(\Gamma=0\)), and the Degasperis-Procesi equation (\(b=3\) and \(c_0=\Gamma=0\)). The Painleve analysis shows that these sets of parameters are the only possible ones which correspond to integrable systems.
Assuming that \(u_0 \in H^r({\mathbb R})\) with \(r>3/2\), the authors describe the blow up scenaria for different values of \(b\), give sufficient conditions for the global well-posedness on the equation as well as some sufficient conditions for a blow up of a solution, estimating therewith the maximal existence time of a solution in terms of the initial data. It appears that for \(b>1\) we may have globally existing strong solutions as well as blow ups and the behavior of a solution is affected by the shape of the initial data.
The local well-posedness of the inital value problem for \(b\)-equations and the uniqueness and existence of a global weak solution are also established assuming that \(c_0 + \frac{\Gamma}{\alpha^2}=0\) and the initial data meet some sign conditions.

MSC:
35Q53 KdV equations (Korteweg-de Vries equations)
35Q58 Other completely integrable PDE (MSC2000)
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35B40 Asymptotic behavior of solutions to PDEs
35D05 Existence of generalized solutions of PDE (MSC2000)
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