Blow-up for nonlinear dissipative wave equations in Rn

We consider the Cauchy problem {utt−Δu+|ut|m−1ut=|u|p−1u,(t,x)∈(0,∞)×Rn,u(0,x)=u0(x),ut(0,x)=v0(x), for 1⩽m0, λ⩾0 there exist infinitely many data u0,v0 in the energy space such that the initial energy E(0)=λ, the gradient norm ‖∇u0‖2=α, and the solution of the above Cauchy problem blows up in finite time.