Global nonexistence for a semilinear wave equation with nonlinear boundary dissipation

In this paper, we consider the following problem { y t t − y x x + y t = | y | p − 1 y , ( x , t ) ∈ ( 0 , L ) × ( 0 , T ) , y ( 0 , t ) = 0 , t ∈ ( 0 , T ) , y x ( L , t ) + y ( L , t ) + | y t ( L , t ) | m − 1 y t ( L , t ) = 0 , t ∈ ( 0 , T ) , y ( x , 0 ) = y 0 ( x ) , y t ( x , 0 ) = y 1 ( x ) , x ∈ ( 0 , L ) , where ( 0 , L ) is a bounded open interval in R , p > 1 and m ⩾ 1 . We are interested in the interaction between the boundary damping | y t ( L , t ) | m − 1 y t ( L , t ) and the interior source | y ( t ) | p − 1 y ( t ) . Under some appropriate assumptions on the initial data, two blow-up results with positive initial energy are established. Furthermore, we obtain that the solutions blow up if p or | y 0 ( L ) | is sufficiently large.