On the nonlinear wave equation utt−B(∥ux∥2)uxx=f(x,t,u,ux,ut,∥ux∥2) associated with the mixed homogeneous conditions Original Research Article

In this paper, we consider the following nonlinear wave equation: (1) utt−B(∥ux∥2)uxx=f(x,t,u,ux,ut,∥ux∥2),x∈(0,1),00,B1∈CN(R+),B1⩾0,f∈CN+1([0,1]×R+×R3×R+) and f1∈CN([0,1]×R+×R3×R+) we obtain from the following equation: utt−[B(∥ux∥2)+εB1(∥ux∥2)]uxx=f(x,t,u,ux,ut,∥ux∥2)+εf1(x,t,u,ux,ut,∥ux∥2) associated to (2), (3) a weak solution uε(x,t) having an asymptotic expansion of order N+1 in ε, for ε sufficiently small.