A nonlinear dynamical system on the set of Laguerre entire functions

A nonlinear modification of the Cauchy problem Dtf(t, z) = θDzf(t, z) + zD 2 zf(t, z), t ∈ IR+ = [0,+∞), z ∈C, θ ≥ 0, f(0, z) = g(z) ∈ L is considered. The set L consists of Laguerre entire functions, which one obtains as a closure of the set of polynomials having real nonpositive zeros only in the topology of uniform convergence on compact subsets of C. The modification means that the time half-line IR+ is divided onto the intervals In = [(n − 1)τ, nτ ], n ∈ IN , τ > 0, and on each In the evolution is to be described by the above equation but at the endpoints the function f(t, z) is changed: f(nτ, z) → [