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 equationutt+k1∇4u+k2∇4ut+∇2g(∇2u)=0,(x,t)∈Ω×(0,T),u=0,∇2u=0,(x,t)∈∂Ω×(0,T),u(x,0)=u0(x),ut(x,0)=u1(x),x∈Ω⊂R3 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.