Thresholds for low regularity solutions to wave equations with structural damping

We study the asymptotic behavior of solutions to wave equations with the structural damping termutt−Δu+Δ2ut=0,u(0,x)=u0(x),ut(0,x)=u1(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 [2] 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.