LAUSR

Abstract We study the Cauchy problem for the strong dissipative fifth-order KdV equations (0.1) ∂ t u + β 1 ∂ x 5 u + β 2 ∂ x 4 u = c 1 u ∂ x u + c 2 u 2 ∂ x u + b 1 ∂ x u ∂ x 2 u + b 2 u ∂ x 3 u , x ∈ R , t ∈ R + , u ( 0 , x ) = u 0 ( x ) ∈ H s ( R ) . We show that the Cauchy problem (0.1) is locally well-posed in H s ( R ) for any s ≥ 0 . Moreover, as u 0 ∈ H 2 ( R ) , b 1 = 2 b 2 , c 2 = 0 and β 2 → 0 , we prove that the global solution to (0.1) converges to the global weak solution of the fifth-order KdV equations (0.2) ∂ t u + β 1 ∂ x 5 u = c 1 u ∂ x u + 2 b 2 ∂ x u ∂ x 2 u + b 2 u ∂ x 3 u . This global solution result is consistent with the result by Kenig and Pilod (2015) and Guo et al. (2013), who used short-time structure method. However, we only use the classical viscous disappearing method to get the global solution to (0.2) .