Existence and uniqueness of a solution of a Wentzell problem with nonlinear delays

We consider the following coupled system wave/Wentzell :

u_tt-Δu+μ₁g₁(u_t)+μ₂g₁(u_t(t-τ))=0, in Ω×(0,∞),

v_tt+∂_νu-Δ_Tv+μ₁′g₂(v_t)+μ₂′g₂(v_t(t-τ))=0, on Γ₁×(0,∞),

u=v, on Γ×(0,∞),

u=0, on Γ₀×(0,∞),

where Ω is a bounded domain in Rⁿ(n≥2), with smooth boundary Γ divided into two closed and disjoint subsets Γ₀ and Γ₁. We denote by ∇_T the tangential gradient on Γ, by Δ_T the tangential Laplacian on Γ and by ∂_ν the normal derivative where ν represents the unit outward normal to Γ.
μ₁, μ₂, μ₁′ and μ₂′ are positive real numbers, the two functions g₁(u_t(t-τ)) and g₂(v_t(t-τ)) describe the delays on the nonlinear frictional dissipations g₁(u_t) and g₂(v_t), on Ω and Γ₁, respectively, τ>0 is a time delay.

we will prove that the above problem is well-posed by proving the existence and uniqueness of a solution using the Faedo-Galerkin method.