On the Cauchy problem for the strong dissipative fifth-order KdV equations

Abstract

We study the Cauchy problem for the strong dissipative fifth-order KdV equations (0.1)∂tu+β1∂x5u+β2∂x4u=c1u∂xu+c2u2∂xu+b1∂xu∂x2u+b2u∂x3u,x∈R,t∈R+,u(0,x)=u0(x)∈Hs(R).We show that the Cauchy problem (0.1) is locally well-posed in Hs(R) for any s≥0. Moreover, as u0∈H2(R), b1=2b2, c2=0 and β2→0, we prove that the global solution to (0.1) converges to the global weak solution of the fifth-order KdV equations (0.2)∂tu+β1∂x5u=c1u∂xu+2b2∂xu∂x2u+b2u∂x3u.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).