  In this paper we employ the Renormalization Group (RG) method to study the
long-time asymptotics of a class of nonlinear integral equations with a
generalized heat kernel and with time-dependent coefficients. The
nonlinearities are classified and studied according to its role in the
asymptotic behavior. Here we prove that adding nonlinear perturbations
classified as irrelevant, the behavior of the solution in the limit $t
\to\infty$ remains unchanged from the linear case. In a companion paper, we
will include a type of nonlinearities called marginal and we will show that, in
this case, the large time limit gains an extra logarithmic decay factor.
