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On ill-posedness for the generalized BBM equation

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  • We consider the Cauchy problem associated to the generalized Benjamin-Bona-Mahony (BBM) equation for given data in the $L^2$-based Sobolev spaces. Depending on the order of nonlinearity and dispersion, we prove that the Cauchy problem is ill-posed for data with lower order Sobolev regularity. We also prove that, in certain range of the Sobolev regularity, even if the solution exists globally in time, it fails to be smooth.
    Mathematics Subject Classification: Primary: 35Q53; Secondary: 35A01.


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