Within the quantum mechanical treatment of the decay problem one finds that at late times $t$ the survival probability of an unstable state cannot have the form of an exponentially decreasing function of time $t$ but it has an inverse power-like form. This is a general property of unstable states following from basic principles of quantum theory. The consequence of this property is that in the case of false vacuum states the cosmological constant becomes dependent on time: $Lambda - Lambda_{text{bare}}equiv Lambda(t) -Lambda_{text{bare}} sim 1/t^{2}$. We construct the cosmological model with decaying vacuum energy density and matter for solving the cosmological constant problem and the coincidence problem. We show the equivalence of the proposed decaying false vacuum cosmology with the $Lambda(t)$ cosmologies (the $Lambda(t)$CDM models). The cosmological implications of the model of decaying vacuum energy (dark energy) are discussed. We constrain the parameters of the model with decaying vacuum using astronomical data. For this aim we use the observation of distant supernovae of type Ia, measurements of $H(z)$, BAO, CMB and others. The model analyzed is in good agreement with observation data and explain a small value of the cosmological constant today.
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