Energy decay for wave equations with nonlinear damping and space-time coefficient in Rn

In this paper, we consider the following nonlinear wave equation u t t − Δ u + a ( t , x ) | u t | m − 1 u t + | u | p − 1 u = 0 , t > 0 , x ∈ R n u ( 0 , x ) = u 0 ( x ) , u t ( 0 , x ) = u 1 ( x ) x ∈ R n and obtain optimal energy decay estimates under suitable assumptions on the space-time nonlinear damping coefficient a , where m satisfies 1 < m ⩽ n + 2 n − 2 . We establish a new Liapunov type function to obtain energy decay estimates in the case of nonlinear damping instead of the Kormonik lemma.