Homogeneous initial–boundary value problem of the Rosenau equation posed on a finite interval

Abstract

This paper considers the IBVP of the Rosenau equation { ∂ t u + ∂ t ∂ x 4 u + ∂ x u + u ∂ x u = 0 , x ∈ ( 0 , 1 ) , t > 0 , u ( 0 , x ) = u 0 ( x ) u ( 0 , t ) = ∂ x 2 u ( 0 , t ) = 0 , u ( 1 , t ) = ∂ x 2 u ( 1 , t ) = 0 . It is proved that this IBVP has a unique global distributional solution u ∈ C ( [ 0 , T ] ; H s ( 0 , 1 ) ) as initial data u 0 ∈ H s ( 0 , 1 ) with s ∈ [ 0 , 4 ] . This is a new global well-posedness result on IBVP of the Rosenau equation with Dirichlet boundary conditions.