Dispersive mixed-order systems in $L^p$-Sobolev spaces and application to the thermoelastic plate equation

TL;DR

This paper investigates dispersive mixed-order pseudodifferential systems in $L^p$-Sobolev spaces, revealing conditions for semigroup generation and applying findings to thermoelastic plate equations with different heat conduction laws.

Contribution

It establishes new conditions under which dispersive mixed-order systems generate semigroups in $L^p$-Sobolev spaces, especially highlighting the cases $p=2$ or $n=1$.

Findings

Semigroup generation depends on $p=2$ or $n=1$ in $L^p$-Sobolev spaces.

Application to thermoelastic plate equations with Fourier and Maxwell-Cattaneo laws.

Weak quasi-hyperbolicity condition suffices for semigroup generation.

Abstract

We study dispersive mixed-order systems of pseudodifferential operators in the setting of $L^p$-Sobolev spaces. Under the weak condition of quasi-hyperbolicity, these operators generate a semigroup in the space of tempered distributions. However, if the basic space is a tuple of $L^p$-Sobolev spaces, a strongly continuous semigroup is in many cases only generated if $p=2$ or $n=1$. The results are applied to the linear thermoelastic plate equation inertial term and with Fourier's or Maxwell-Cattaneo's law of heat conduction.