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In this article, we study the blow‐up of the damped wave equation in the  scale‐invariant case  and in the presence of two nonlinearities. More precisely, we consider the following equation:u    t  t    −  Δ  u  +    μ    1  +  tu    t    =  |    u    t|    p    +  |  u    |    q    ,    inℝ    N    ×  [  0  ,  ∞  )  ,    with small initial data.  For   μ  <    N  (  q  −  1  )    2and   μ  ∈ (0,  μ  ∗ ) , where   μ  ∗  > 0  is depending on the nonlinearties' powers and the space dimension (  μ  ∗   satisfies   (  q  −  1  )    (  N  +  2    μ    ∗    −  1  )  p  −  2    =  4  ), we prove that the wave equation, in this case, behaves like the one without dissipation (  μ  = 0 ). Our result completes the previous studies in the case where the dissipation is given byμ(  1  +  t  )    βu    t    ;    β  >  1  , where, contrary to what we obtain in the present work, the effect of the damping is not significant in the dynamics. Interestingly, in our case, the influence of the damping termμ    1  +  tu    tis important.

Blow‐up for wave equation with the scale‐invariant damping and combined nonlinearities