Please login first
Existence and uniqueness of a solution of a Wentzell problem with nonlinear delays
1  Mathematics (Partial Differential Equations)
Academic Editor: Alberto Cabada


We consider the following coupled system wave/Wentzell :

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

vtt+∂νu-ΔTv+μ₁′g₂(vt)+μ₂′g₂(vt(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 ΔT 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(ut(t-τ)) and g₂(vt(t-τ)) describe the delays on the nonlinear frictional dissipations g₁(ut) and g₂(vt), 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.

Keywords: Wave equation, Wentzell dynamical boundary conditions, internal and boundary nonlinear delay terms, Galerkin approximation scheme.