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

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September 2009, 8(5): 1521-1539. doi: 10.3934/cpaa.2009.8.1521

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

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

Received March 2008 Revised October 2008 Published April 2009

Abstract
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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: unconditional well-posedness., Local well-posedness.

Mathematics Subject Classification: Primary: 35B30, 35Q55; Secondary: 35Q7.

Citation: Luiz Gustavo Farah. Local solutions in Sobolev spaces and unconditional well-posedness for the generalized Boussinesq equation. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1521-1539. doi: 10.3934/cpaa.2009.8.1521

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