Local existence and global nonexistence theorems for a damped nonlinear hyperbolic equation

The existence and uniqueness of the local generalized solution to the initial boundary value problem for the three-dimensional damped nonlinear hyperbolic equation u t t + k 1 ∇ 4 u + k 2 ∇ 4 u t + ∇ 2 g ( ∇ 2 u ) = 0 , ( x , t ) ∈ Ω × ( 0 , T ) , u = 0 , ∇ 2 u = 0 , ( x , t ) ∈ ∂ Ω × ( 0 , T ) , u ( x , 0 ) = u 0 ( x ) , u t ( x , 0 ) = u 1 ( x ) , x ∈ Ω ⊂ R 3 are proved. The paper arrives at some sufficient conditions for blow up of the solutions in finite time by two methods. An example is given.