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

TL;DR

This paper investigates how low regularity of nonlinear terms affects the solution regularity of semilinear heat and Schr"odinger equations, revealing ill-posedness and optimal regularity results in certain function spaces.

Contribution

It introduces two methods to analyze the impact of nonlinear term regularity, providing new ill-posedness and regularity results for these equations.

Findings

Optimal regularity results in H"older spaces for the heat equation.

Ill-posedness in certain Sobolev spaces for NLS depending on nonlinearity.

Ill-posedness of nonlinear heat equation in H^s for small lpha and large N.

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 $s\ge 0$.