## Advances in Differential Equations

- Adv. Differential Equations
- Volume 20, Number 5/6 (2015), 433-462.

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

Mohamed Kainane Mezadek and Michael Reissig

#### 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.

#### Article information

**Source**

Adv. Differential Equations Volume 20, Number 5/6 (2015), 433-462.

**Dates**

First available in Project Euclid: 30 March 2015

**Permanent link to this document**

https://projecteuclid.org/euclid.ade/1427744012

**Mathematical Reviews number (MathSciNet)**

MR3327703

**Zentralblatt MATH identifier**

1332.35382

**Subjects**

Primary: 35G10: Initial value problems for linear higher-order equations 35Q41: Time-dependent Schrödinger equations, Dirac equations

#### Citation

Mezadek, Mohamed Kainane; Reissig, Michael. Qualitative properties of solutions to structurally damped $\sigma$-evolution models with time increasing coefficient in the dissipation. Adv. Differential Equations 20 (2015), no. 5/6, 433--462. https://projecteuclid.org/euclid.ade/1427744012