Semilinear structural damped waves

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We study the Cauchy problem for the semilinear structural damped wave equation with source term utt−△u+μ(−Δ)σut=f(u),u(0,x)=u0(x),ut(0,x)=u1(x),

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A new phenomenon in the critical exponent for structurally damped semi-linear evolution equations (2017, unknown): “In the case of structurally damped waves with power nonlinearity | u | p , i.e. σ = 1 and δ ∈ (0 , 1] in (1), the first global existence result has been obtained by the first author and M. Reissig [15] in low space dimension, by using energy estimates.”
On the global existence and blowup of smooth solutions of 3-D compressible Euler equations with time-depending damping (2015, background): “ difﬁcult, however, to apply these microlocal estimates to attack the quasilinear problem (1.5). (In general, those microlocal estimates are useful when treating semilinear damped wave equations, see [8,9] and references therein.) Remark 1.2. For the 1-D Burgers equation with damping term ∂tw+w∂x”