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May/June 2015 Qualitative properties of solutions to structurally damped $\sigma$-evolution models with time increasing coefficient in the dissipation
Mohamed Kainane Mezadek, Michael Reissig
Adv. Differential Equations 20(5/6): 433-462 (May/June 2015).

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.

Citation

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Mohamed Kainane Mezadek. Michael Reissig. "Qualitative properties of solutions to structurally damped $\sigma$-evolution models with time increasing coefficient in the dissipation." Adv. Differential Equations 20 (5/6) 433 - 462, May/June 2015.

Information

Published: May/June 2015
First available in Project Euclid: 30 March 2015

zbMATH: 1332.35382
MathSciNet: MR3327703

Subjects:
Primary: 35G10, 35Q41

Rights: Copyright © 2015 Khayyam Publishing, Inc.

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Vol.20 • No. 5/6 • May/June 2015
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