Local solutions for a coupled system of Kirchhoff type

Abstract

We investigate the existence of local solutions of the following coupled system of Kirchhoff equations subject to nonlinear dissipation on the boundary: ( ∗ ) | u ″ − M 1 ( t , ‖ u ( t ) ‖ 2 , ‖ v ( t ) ‖ 2 ) △ u = 0 in Ω × ( 0 , ∞ ) , v ″ − M 2 ( t , ‖ u ( t ) ‖ 2 , ‖ v ( t ) ‖ 2 ) △ v = 0 in Ω × ( 0 , ∞ ) , u = 0 , v = 0 on Γ 0 × ] 0 , ∞ [ , ∂ u ∂ ν + δ 1 h 1 ( u ′ ) = 0 on Γ 1 × ] 0 , ∞ [ , ∂ u ∂ ν + δ 2 h 2 ( u ′ ) = 0 on Γ 1 × ] 0 , ∞ [ . Here { Γ 0 , Γ 1 } is an appropriate partition of the boundary Γ of Ω and ν ( x ) , the outer unit normal vector at x ∈ Γ 1 . 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 (∗).