Local solutions in Sobolev spaces and unconditional well-posedness for the generalized Boussinesq equation

Luiz Gustavo Farah

doi:10.3934/cpaa.2009.8.1521

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2009, Volume 8, Issue 5: 1521-1539. Doi: 10.3934/cpaa.2009.8.1521

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Local solutions in Sobolev spaces and unconditional well-posedness for the generalized Boussinesq equation

Luiz Gustavo Farah^1,

1.
Instituto de Matemática Pura e Aplicada (IMPA), Estrada Dona Castorina 110, Rio de Janeiro, 22460-320

Received: February 29, 2008

Revised: September 30, 2008

Published: August 2009

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Abstract

We study the local well-posedness of the initial-value problem for the nonlinear generalized Boussinesq equation with data in $H^s(\mathbb R^n) \times H^s(\mathbb R^n)$, $s\geq 0$. Under some assumption on the nonlinearity $f$, local existence results are proved for $H^s(\mathbb R^n)$-solutions using an auxiliary space of Lebesgue type. Furthermore, under certain hypotheses on $s$, $n$ and the growth rate of $f$ these auxiliary conditions can be eliminated.

Keywords:

Local well-posedness,
unconditional well-posedness.

Mathematics Subject Classification: Primary: 35B30, 35Q55; Secondary: 35Q72.

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