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Existence, decay and blow-up for solutions to the sixth-order generalized Boussinesq equation

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  • We study the existence, the decay and the blow-up of solutions to the Cauchy problem for the multi-dimensional generalized sixth-order Boussinesq equation: $$ u_{tt} - \Delta u - \Delta^{2} u- \mu \Delta ^{3} u = \Delta f(u),\; t>0, \; x \in {\mathbb{R}^{n}}, n \geq 1, $$ where $ f(u)= \gamma |u|^{p-1}u, \; \gamma \in \mathbb{R}, \; p \geq 2, \; \mu > 1/4$. We find two global existence results for appropriate initial data when $n$ verifies $1 \leq n\leq 4(p+1)/(p-1).$ On the other hand we show that if $\mu= 1/3$ and $p>13/2$, then the solution with small initial data decays in time. A blow up in finite time result is also obtained for appropriate initial data when $n$ verifies $1 \leq n\leq 4(p+1)/(p-1).$
    Mathematics Subject Classification: 35A01, 35B40, 35B44, 35Q35.

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