具阻尼项的“坏”的Boussinesq型方程的Cauchy问题(英文)

Cauchy Problem for the "Bad" Boussinesq-Type Equation with Damping Term

研究一类具阻尼项的"坏"的Boussinesq型方程utt-uxx-2kuxxt-αuxxxx=β(un)xx的Cauchy问题,其中k,α为大于零的实数,β是实数,n≥2是整数。在关于初值的适当假设下,证明了Cauchy问题存在一个整体光滑解u∈C∞((0,T];H∞(R))∩C([0,T];H1(R))∩C1([0,T];H-1(R))对任何T>0。

Abstract The Cauehy problem for the ＂bad＂ Boussinesq-type equation with damping term uu - uxx - 2kuxxt - auxxx = β（ u^n ）xx is studied. Where k, α and β are real numbers, with k 〉 0, a 〉 0, and n≥ 2 is an integer. It proves that for any T 〉 0 the Cauchy problem admits a global smooth solution u∈C^∞（（0,T];H^∞（R））∩C（[0,T];H^1（R））∩C^1（[0,T];H^-1（R）） under suitable assumptions on the initial data.