Exponential decay of the viscoelastic wave equation of Kirchhoff type with a nonlocal dissipation

The following viscoelastic wave equation of Kirchhoff type with nonlinear and nonlocal damping u(tt) - psi(parallel to del u parallel to(2)(2)) Delta u - alpha Delta u(t) + integral(t)(0) g(t - tau)Delta u(tau)d tau + M(parallel to del u parallel to(2)(2)) u(t) = f(u), where M(r) is a C-1([0, infinity)) -function satisfying M(r) >= m(1) > 0 for r >= 0, is considered in a bounded domain Omega of R-N. The existence of global solutions and decay rates of the energy are proved.