  A nonlinear modification of a parabolic Cauchy problem for entire functions
of a single complex variable is considered. The modification means that the
time half-line is divided onto the intervals of equal length and on each such
interval the evolution is to be described by the mentioned equation but at the
endpoints the function is changed in a nonlinear way. If the initial function
is chosen in the set of Laguerre entire functions, then the solution of the
problem remains in this set. The Laguerre entire functions are obtained as
uniform limits on compact subsets of the complex plane of the polynomials
having real nonpositive zeros only. It is shown that the asymptotic properties
of the solution change considerably when interval length reaches a threshold
value. Certain applications, including limit theorems for weakly and strongly
dependent random vectors, are given.
