Global solutions for a nonlinear Kirchhoff type equation with viscosity

In this paper we consider the existence and asymptotic behavior of solutions of the following nonlinear Kirchhoff type problem $ u_{t t} - M ( \int_\Omega | \nabla u |^2 dx ) \Delta u - \delta \Delta u_t = \mu | u |^{p-2} u $ in $ \Omega \times ]0, \infty [ $ where $ M(s) = $ ⎧ $ a - bs $ for $ s \in [0, a/b [$, ⎨ ⎩ $ 0, $ for $ s \in [ a/b, + \infty ] $. If the initial energy is appropriately small, we derive the global existence theorem and its exponential decay.