Local solutions for a coupled system of Kirchhoff type

We investigate the existence of local solutions of the following coupled system of Kirchhoff equations subject to nonlinear dissipation on the boundary: equation(∗∗)|u″−M1(t,‖u(t)‖2,‖v(t)‖2)△u=0in Ω×(0,∞),v″−M2(t,‖u(t)‖2,‖v(t)‖2)△v=0in Ω×(0,∞),u=0,v=0on Γ0×]0,∞[,∂u∂ν+δ1h1(u′)=0on Γ1×]0,∞[,∂u∂ν+δ2h2(u′)=0on Γ1×]0,∞[. Here {Γ0,Γ1}{Γ0,Γ1} is an appropriate partition of the boundary ΓΓ of ΩΩ and ν(x)ν(x), the outer unit normal vector at x∈Γ1x∈Γ1.By applying the Galerkin method with a special basis for the space where lie the approximations of the initial data, we obtain local solutions of the initial-boundary value problem for (∗).