On a nonlinear wave equation associated with the boundary conditions involving convolution

The paper deals with the initial boundary value problem for the linear wave equation equation(1){utt−∂∂x(μ(x,t)ux)+λut+F(u)=0,00μ(x,t)≥μ0>0, a.e. (x,t)∈QT(x,t)∈QT; the function FF continuous, ∫0zF(s)ds≥−C1z2−C1′, for all z∈Rz∈R, with C1C1, C1′>0 are given constants and some other conditions, we prove that, the problem (1) has a unique weak solution uu. The proof is based on the Faedo–Galerkin method associated with the weak compact method. In Part 2 we prove that the unique solution uu belongs to H2(QT)∩H2(QT)∩L∞(0,T;H2)∩C0(0,T;H1)∩L∞(0,T;H2)∩C0(0,T;H1)∩C1(0,T;L2)C1(0,T;L2), with ut∈L∞(0,T;H1)ut∈L∞(0,T;H1), utt∈L∞(0,T;L2)utt∈L∞(0,T;L2), if we assume (u0,u1)∈H2×H1, F∈C1(R)F∈C1(R) and some other conditions. In Part 3, with F∈CN+1(R)F∈CN+1(R), N≥2N≥2, we obtain an asymptotic expansion of the solution uu of the problem (1) up to order N+1N+1 in a small parameter λλ. Finally, in Part 4, we prove that the solution uu of this problem is stable with respect to the data (λλ, μμ, g0g0, g1g1, k0k0, k1k1).