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

Consider the initial–boundary value problem for the nonlinear wave equation equation(1) View the MathML source{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) View the MathML sourceP(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, Turn MathJax on 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), View the MathML sourceg∈Lq0′(0,T), λ=1λ=1, KK, K0≥0K0≥0; pp, p0p0, q0q0, p1p1, q1≥2q1≥2, q>1q>1, View the MathML sourceq0′=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.