Existence and asymptotic expansion for a nonlinear wave equation associated with nonlinear boundary conditions

Consider the initial–boundary value problem for the nonlinear wave equationequation(1){utt−∂∂x(μ(x,t)ux)+f(u,ut)=F(x,t),00μ(x,t)≥μ0>0, μt∈L1(0,T;L∞)μt∈L1(0,T;L∞), μt(x,t)≤0μt(x,t)≤0, a.e. (x,t)∈QT(x,t)∈QT; K0K0, K1≥0K1≥0; p0p0, q0q0, p1p1, q1≥2q1≥2, q0′=q0q0−1, the function ff supposed to be continuous with respect to two variables and nondecreasing with respect to the second variable and some others, we prove that the problem  and  has a weak solution (u,P)(u,P). If, in addition, k∈W1,1(0,T)k∈W1,1(0,T), p0p0, p1∈{2}∪[3,+∞)p1∈{2}∪[3,+∞) and some other conditions, then the solution is unique. The proof is based on the Faedo–Galerkin method and the weak compact method associated with a monotone operator. For the case of q0=q1=2;p0,p1≥2q0=q1=2;p0,p1≥2, in Part 2 we prove that the unique solution (u,P)(u,P) belongs to (L∞(0,T;H2)∩C0(0,T;H1)∩C1(0,T;L2))×H1(0,T)(L∞(0,T;H2)∩C0(0,T;H1)∩C1(0,T;L2))×H1(0,T), with ut∈L∞(0,T;H1)ut∈L∞(0,T;H1), utt∈L∞(0,T;L2)utt∈L∞(0,T;L2), u(0,⋅)u(0,⋅), u(1,⋅)∈H2(0,T)u(1,⋅)∈H2(0,T), if we assume (u0,u1)∈H2×H1, f∈C1(R2)f∈C1(R2) and some other conditions. Finally, in Part 3, with q0=q1=2q0=q1=2; p0p0, p1≥N+1p1≥N+1, f∈CN+1(R2)f∈CN+1(R2), N≥2N≥2, we obtain an asymptotic expansion of the solution (u,P)(u,P) of the problem  and  up to order N+1N+1 in two small parameters K0K0, K1K1.