Blow-up for nonlinear dissipative wave equations in [formula omitted

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We consider the Cauchy problem { u t t − Δ u + | u t | m − 1 u t = | u | p − 1 u , ( t , x ) ∈ ( 0 , ∞ ) × R n , u ( 0 , x ) = u 0 ( x ) , u t ( 0 , x ) = v 0 ( x ) , for 1 ⩽ m < p , p < n / ( n − 2 ) for n ⩾ 3 . We prove that for any given numbers α > 0 , λ ⩾ 0 there exist infinitely many data u 0 , v 0 in the energy space such that the initial energy E ( 0 ) = λ , the gradient norm ‖ ∇ u 0 ‖ 2 = α , and the solution of the above Cauchy problem blows up in finite time.