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The main purpose of this paper is to study the following semi‐linear structurally damped wave equation with nonlinearity of derivative type:u    t  t    −  Δ  u  +  μ    (  −  Δ  )    σ  /  2u    t    =  |    u    t|    p    ,    u  (  0  ,  x  )  =    u    0    (  x  )  ,u    t    (  0  ,  x  )  =    u    1    (  x  )  ,    with  μ >0,  n  ≥ 1,  σ ∈(0,2], and  p >1. In particular, we would like to prove the nonexistence of global weak solutions by using a new test function and suitable sign assumptions on the initial data in both the subcritical case and the critical case.

Critical exponent for semi‐linear structurally damped wave equation of derivative type