Weak solutions to a nonlinear variational sine–Gordon equation

This paper considers a nonlinear variational sine–Gordon equation u t t − c ( u ) [ c ( u ) u x ] x + λ 2 2 sin ( 2 u ) = 0 which describes the motion of long waves on a neutral dipole chain in the continuum limit and a few other physical phenomena. We establish the global existence of dissipative weak solutions to its Cauchy problem for initial data in the space H 1 × L 2 by using the Young measure theory in the setting of L p spaces.