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

Consider the initial–boundary value problem for the nonlinear wave equation equation(1) View the MathML source{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, View the MathML sourceq0′=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 (1) and (2) 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 View the MathML source(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 (1) and (2) up to order N+1N+1 in two small parameters K0K0, K1K1.