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Existence Theory for the Boussinesq Equation in Modulation Spaces

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Abstract

In this paper we study the Cauchy problem for the generalized Boussinesq equation with initial data in modulation spaces \(M^{s}_{p^\prime ,q}(\mathbb {R}^n),\) \(n\ge 1.\) After a decomposition of the Boussinesq equation in a \(2\times 2\)-nonlinear system, we obtain the existence of global and local solutions in several classes of functions with values in \( M^s_{p,q}\times D^{-1}JM^s_{p,q}\)-spaces for suitable p, q and s,  including the special case \(p=2,q=1\) and \(s=0.\) Finally, we prove some results of scattering and asymptotic stability in the framework of modulation spaces.

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Acknowledgements

The second author was supported by the Vicerrectoría de Investigación y Extensión-UIS and Fondo Nacional de Financiamiento para la Ciencia, la Tecnología y la Innovación Francisco José de Caldas, contrato Colciencias FP 44842-157-2016.

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Authors and Affiliations

  1. Departamento de Matemáticas y Estadística, Universidad de Córdoba, A.A. 354, Montería, Colombia

    Carlos Banquet

  2. Universidad Industrial de Santander, Escuela de Matemáticas, A.A. 678, Bucaramanga, Colombia

    Élder J. Villamizar-Roa

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  1. Carlos Banquet
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  2. Élder J. Villamizar-Roa
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Correspondence to Carlos Banquet.

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Banquet, C., Villamizar-Roa, É.J. Existence Theory for the Boussinesq Equation in Modulation Spaces. Bull Braz Math Soc, New Series 51, 1057–1082 (2020). https://doi.org/10.1007/s00574-019-00188-3

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  • DOI: https://doi.org/10.1007/s00574-019-00188-3

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