On the nonlinear wave equation with the mixed nonhomogeneous conditions: Linear approximation and asymptotic expansion of solutions

In this paper, we consider the following nonlinear wave equation equation(1){utt−∂∂x(μ(u)ux)=f(x,t,u,ux,ut),00μ(z)≥μ0>0, μ1(z)≥0μ1(z)≥0, for all z∈Rz∈R, and g∈C3(R+)g∈C3(R+), f∈CN+1([0,1]×R+×R3)f∈CN+1([0,1]×R+×R3), f1∈CN([0,1]×R+×R3)f1∈CN([0,1]×R+×R3), a weak solution uε1,ε2(x,t)uε1,ε2(x,t) having an asymptotic expansion of order N+1N+1 in two small parameters ε1ε1, ε2ε2 is established for the following equation associated to (1)2,3: equation(2)utt−∂∂x([μ(u)+ε1μ1(u)]ux)=f(x,t,u,ux,ut)+ε2f1(x,t,u,ux,ut).