On ill-posedness for the generalized BBM equation

Xavier Carvajal; Mahendra Panthee

doi:10.3934/dcds.2014.34.4565

Article Contents

2014, Volume 34, Issue 11: 4565-4576. Doi: 10.3934/dcds.2014.34.4565

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On ill-posedness for the generalized BBM equation

Xavier Carvajal^1, and
Mahendra Panthee^2,

1.
Instituto de Matemática - UFRJ Av. Horácio Macedo, Centro de Tecnologia, Cidade Universitária, Ilha do Fundão, 21941-972 Rio de Janeiro, RJ
2.
Departamento de Matemática, Universidade Estadual de Campinas (UNICAMP), Rua Sergio Buarque de Holanda, 651, 13083-859, Campinas, SP

Received: September 30, 2013

Revised: November 30, 2013

Published: October 2014

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Abstract

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.

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Mathematics Subject Classification: Primary: 35Q53; Secondary: 35A01.

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