Qualitative properties of solutions to structurally damped σevolution models with time increasing coefficient in the dissipation

The goal of this paper is to study qualitative properties of solutions to the Cauchy problem for structurally damped σevolution models u_tt + (-δ)^σu + b(t)(-δ)δu_t = 0; u(0; x) = u₀(x); u_t(0; x) = u¹(x); where σ > 1, δ (0, σ), and the dissipation coecient b = b(t) is a timedependent and strictly increasing positive function. On the one hand, we are interested in L^p - L^q estimates for the energies of higher order. On the other hand, we are interested in Gevrey smoothing properties of solutions. Finally, we prove the optimality of decay by using scaleinvariant models. The main tool of our considerations is a related WKB-analysis.