It is well-known that the Laguerre entire functions are obtained as uniform limits on compact subsets of \(\mathbb C\) of the sequences of polynomials which have only real nonpositive zeros. In [Complex Variables, Theory Appl. 44, No. 3, 225–244 (2001; Zbl 1037.46022)], the authors considered the set \(L\) of such functions and described it in the framework of locally convex spaces of exponential type entire functions. Among other things, it was proved that the Cauchy problem \[ \begin{cases} \frac{\partial f(t,z)}{\partial t}=\theta\frac{\partial f(t,z)}{\partial z}+z\frac{\partial^2f(t,z)}{\partial z^2},\, t\in\mathbb R^+,\, z\in\mathbb C,\\ f(0,z)=g(z),\, g\in L,\, \theta\geq 0,\end{cases} \] has a unique solution in \(L\), at least for \(t\) small enough.
In the present paper, the authors consider a nonlinear modification of this problem and prove new results. This is an essential contribution in the subject.