On a Model Connected with the Kirchhoff Equation

We prove a global existence result in suitable Sobolev spaces for the solution of the nonlinear Cauchy problem ∂ttu t x − 1 + Rn dy K x − y ∇yu t y 2 xu t x = 0 u 0 x = u0 x ∂tu 0 x = u1 x >0 where n > 3 is the space dimension, the initial data are C∞ with compact support in Rn, and K z is a positive smooth function rapidly decreasing as z → ∞. Our approach is based on the energy estimates combined with the classical Von Wahl inequalities.