We investigate the existence and uniqueness of solutions of the following equation of hyperbolic type with a strong dissipation: u t t ( t , x ) − ( α + β ( ∫ Ω | ∇ u ( t , y ) | 2 d y ) γ ) Δ u ( t , x )                                                                 − λ Δ u t ( t , x ) + μ | u ( t , x ) | q − 1 u ( t , x ) = 0 ,           x ∈ Ω , t ≥ 0                         u ( 0 , x ) = u 0 ( x ) ,                     u t ( 0 , x ) = u 1 ( x ) ,             x ∈ Ω ,     u | ∂ Ω = 0 , where 1, \lambda > 0, \mu \in \mathbb{R}, \alpha, \beta \ge 0, \alpha + \beta > 0$"> q > 1 , λ > 0 , μ ∈ ℝ , α , β ≥ 0 , α + β > 0 , and Δ is the Laplacian in ℝ N .