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

The paper deals with the initial boundary value problem for the linear wave equation equation(1) View the MathML source{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, View the MathML source∫0zF(s)ds≥−C1z2−C1′, for all z∈Rz∈R, with C1C1, View the MathML sourceC1′>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 View the MathML source(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).