Exponential Decay of Energy for Some Nonlinear Hyperbolic Equations with Strong Dissipation

The initial boundary value problem for a class of hyperbolic equations with strong dissipative term u(tt) - Sigma(n)(t=1)(partial derivative/partial derivative x(i))(|partial derivative u/partial derivative x(i)|(p-2)(partial derivative u/partial derivative x(i)))-a Delta u(t) = b|u|(r-2)u in a bounded domain is studied. The existence of global solutions for this problem is proved by constructing a stable set inW(0)(1,p) (Omega) and showing the exponential decay of the energy of global solutions through the use of an important lemma of V. Komornik.