Qualitative properties of solutions to structurally damped $\sigma$-evolution models with time increasing coefficient in the dissipation

Abstract

The goal of this paper is to study qualitative properties of solutions to the Cauchy problem for structurally damped $\sigma-$evolution models \begin{align*} & u_{tt}+(-\Delta)^\sigma u +b(t)(-\Delta)^\delta u_{t}=0, \\ & u(0,x)=u_{0}(x),\,\,\,\, u_t(0,x)=u_{1}(x), \end{align*} where $\sigma>1$, $\delta \in(0,\sigma)$, and the dissipation coefficient $b=b(t)$ is a time-dependent 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 scale-invariant models. The main tool of our considerations is a related WKB-analysis.