  We study one of the strongest versions of chaos for continuous dynamical
systems, namely the specification property. We extend the definition of
specification property for operators on a Banach space to strongly continuous
one-parameter semigroups of operators, that is, $C_0$-semigroups. In addition,
we study the relationships of the specification property for $C_0$-semigroups
(SgSP) with other dynamical properties: mixing, Devaney's chaos, distributional
chaos and frequent hypercyclicity. Concerning the applications, we provide
several examples of semigroups which exhibit the SgSP with particular interest
on solution semigroups to certain linear PDEs, which range from the hyperbolic
heat equation to the Black-Scholes equation.
