Periodic solutions for wave equations with variable coefficients with nonlinear localized damping

We discuss the existence of periodic solutions to the wave equation with variable coefficients u t t − div ( A ( x ) ∇ u ) + ρ ( x , u t ) = f ( x , t ) with Dirichlet boundary condition. Here ρ ( x , v ) is a function like ρ ( x , v ) = a ( x ) g ( v ) with g ′ ( v ) ⩾ 0 where a ( x ) is nonnegative, being positive only in a neighborhood of a part of the domain.