In that paper, we prove an equivalence between Nash-type inequalities and an exponential decay (in the sense of the definition \ref{defent}) for symmetric submarkovian semigroups. This exponential decay generalizes the notion of spectral gap where this number is replaced by a function. We discuss different formulations of the decay associated to the usual Nash inequality in terms of Lyapunov-type functional. We apply this to different classes of ultracontractive semigroups as well as non-ultracontractive semigroups. In particular, we show that any ultracontractive semigroups always satisfy an exponential decay in the sense of \ref{defent}. We treat different classes of examples, one of them containing the Ornstein-Uhlenbeck-type semigroup and ${\Gamma}^*$-semigroup. We apply our results to fractional powers of non-negative self-adjoint semigroup. We derive a simple criterium on the function charaterizing the exponential decay to deduce ultracontractivity property and relations that must satisfy the ultracontractive bounds an heat kernel of the semigroup.