  The decay rate of metastable states is determined at high temperatures by
thermal activation, whereas at temperatures close to zero quantum tunneling is
relevant. At some temperature $T_{c}$ the transition from classical to
quantum-dominated decay occurs. The transition can be first-order like, with a
discontinuous first derivative of the Euclidean action, or smooth with only a
second derivative developing a jump. In the former case the crossover
temperature $T_{c}$ cannot be calculated perturbatively and must be found as
the intersection point of the Euclidean actions calculated at low and high
temperatures. In this paper we present a sufficient criterion for a first-order
transition in tunneling problems and apply it to the problem of the tunneling
of strings. It is shown that the problem of the depinning of a massive string
from a linear defect in the presence of an arbitrarily strong dissipation
exhibits a first-order transition.
