ON SOLVABILITY OF THE DISSIPATIVE KIRCHHOFF EQUATION WITH NONLINEAR BOUNDARY DAMPING

Abstract. In this paper, we prove the global existence and uniqueness of the dissipative Kirchhoff equation utt −M(‖∇u‖2) △ u+ αut + f(u) = 0 inΩ × [0,∞), u(x, t) = 0 onΓ1 × [0,∞), ∂u ∂ν + g(ut) = 0 on Γ0 × [0,∞), u(x, 0) = u0, ut(x, 0) = u1 inΩ with nonlinear boundary damping by Galerkin approximation benefited from the ideas of Zhang et al. [33]. Furthermore,we overcome some dif-ficulties due to the presence of nonlinear terms M(‖∇u‖2) and g(ut) by introducing a new variables and we can transform the boundary value problem into an equivalent one with zero initial data by argument of compacity and monotonicity. 1