  In this paper we study the Cauchy problem for the semilinear heat and
Schr\"odinger equations, with the nonlinear term $ f ( u ) = \lambda |u|^\alpha
u$. We show that low regularity of $f$ (i.e., $\alpha >0$ but small) limits the
regularity of any possible solution for a certain class of smooth initial data.
We employ two different methods, which yield two different types of results. On
the one hand, we consider the semilinear equation as a perturbation of the ODE
$w_t= f(w)$. This yields in particular an optimal regularity result for the
semilinear heat equation in H\"older spaces. In addition, this approach yields
ill-posedness results for NLS in certain $H^s$ spaces, which depend on the
smallness of $\alpha $ rather than the scaling properties of the equation. Our
second method is to consider the semilinear equation as a perturbation of the
linear equation via Duhamel's formula. This yields in particular that if
$\alpha $ is sufficiently small and $N$ sufficiently large, then the nonlinear
heat equation is ill-posed in $H^s ({\mathbb R}^N ) $ for all $s\ge 0$.
