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