In this paper, we consider the Cauchy problem for a family of
evolution-parabolic coupled systems, which are related to the classical
thermoelastic plate equations containing non-local operators. By using
diagonalization procedure and WKB analysis, we derive representation of
solutions in the phase space. Then, sharp decay properties in a
framework of $L^p-L^q$ are investigated via these
representations. Particularly, some thresholds for the regularity-loss
type decay properties are found.
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