A nonlinear wave equation associated with nonlinear boundary conditions: Existence and asymptotic expansion of solutions

Consider the initial–boundary value problem for the nonlinear wave equation equation(1){utt−uxx+K|u|p−2u+λ|ut|q−2ut=F(x,t),01q>1, KK, λλ are given constants and u0u0, u1u1, FF are given functions, and the unknown function u(x,t)u(x,t) and the unknown boundary value P(t)P(t) satisfy the following nonlinear integral equation equation(2)P(t)=g(t)+K0|u(0,t)|p0−2u(0,t)+|ut(0,t)|q0−2ut(0,t)−∫0tk(t−s)u(0,s)ds, where p0p0, q0≥2q0≥2, K0K0 are given constants and gg, kk are given functions.In this paper, we consider three main parts. In Part 1, under the conditions (u0,u1)∈H1×L2(u0,u1)∈H1×L2, F∈L2(QT)F∈L2(QT), k∈W1,1(0,T)k∈W1,1(0,T), g∈Lq0′(0,T), λ=1λ=1, KK, K0≥0K0≥0; pp, p0p0, q0q0, p1p1, q1≥2q1≥2, q>1q>1, q0′=q0q0−1, we prove a theorem of existence and uniqueness of a weak solution (u,P)(u,P) of problem (1) and (2). 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=2q0=q1=2, pp, qq, p0p0, p1p1≥2≥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 make the assumption that (u0,u1)∈H2×H1(u0,u1)∈H2×H1 and some others. Finally, in Part 3 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 three small parameters KK, λλ, K0K0.