Thresholds for low regularity solutions to wave equations with structural damping

We study the asymptotic behavior of solutions to wave equations with a structural damping term [ u_{tt}-Delta u+Delta^2 u_t=0, qquad u(0,x)=u_0(x), ,,, u_t(0,x)=u_1(x), ] in the whole space. New thresholds are reported in this paper that indicate which of the diffusion wave property and the non-diffusive structure dominates in low regularity cases. We develop to that end the previous authors research in 2019 where they have proposed a threshold that expresses whether the parabolic-like property or the wave-like property strongly appears in the solution to some regularity-loss type dissipative wave equation.