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

Abstract

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 $s\ge 0$.