EXISTENCE, DECAY AND BLOW-UP FOR SOLUTIONS TO THE SIXTH-ORDER GENERALIZED BOUSSINESQ EQUATION

摘要

We study the existence, the decay and the blow-up of solutions to the Cauchy problem for the multi-dimensional generalized sixth-order Boussi-nesq equation: u_(tt) - Δu- Δ~2u - μΔ~3u = Δf(u), t > 0, x ∈ R~n, n ≥ 1, where f(u) = γ|u|~(p~1)u, γ ∈R, p ≥ 2, μ > 1/4. We find two global existence results for appropriate initial data when n verifies 1 ≤ n ≤ 4(p + l)/(p - 1). On the other hand we show that if μ = 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 ≤ n ≤ 4(p + l)/(p - 1).