Blow-up for nonlinear dissipative wave equations in Rn

We consider the Cauchy problem ut t − Δu + |ut |m−1ut = |u|p−1u, (t , x) ∈ (0,∞) ×Rn, u(0,x) = u0(x), ut (0,x) = v0(x), for 1 m 0, 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.  2004 Elsevier Inc. All rights reserved.