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

Non-regularity in H\"older and Sobolev spaces of solutions to the semilinear heat and Schr\"odinger equations