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 $$u_{tt}-\mathrm{\Delta }u+{\mathrm{\Delta }}^2{u}_t=0,u(0,x)=u_0(x),u_t(0,x)=u_1(x),$$ 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 author’s 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.