# Global rough solution for $L^2$-critical semilinear heat equation in the negative Sobolev space

## TL;DR

This paper establishes well-posedness for the $L^2$-critical semilinear heat equation with initial data in negative Sobolev spaces, extending previous results to broader initial conditions including some $L^p$ spaces with $p<2$.

## Contribution

It proves local and global well-posedness for initial data in negative Sobolev spaces, including certain $L^p$ spaces with $p<2$, which was not previously known.

## Key findings

- Existence of a positive constant $oldsymbol{ extit{ extepsilon}_0}$ for well-posedness.
- Well-posedness holds for radial, support-away-from-origin initial data in $oldsymbol{ ext{dot}H^{- extit{ extepsilon}_0}}$.
- Unconditional uniqueness and $L^2$-estimates as $t o 0$ and $t o o ext{infinity}$.

## Abstract

In this paper, we consider the Cauchy global problem for the $L^2$-critical semilinear heat equations $\partial_t h=\Delta h\pm |h|^{\frac4d}h, $ with $h(0,x)=h_0$, where $h$ is an unknown real function defined on $ \R^+\times\R^d$. In most of the studies on this subject, the initial data $h_0$ belongs to Lebesgue spaces $L^p(\R^d)$ for some $p\ge 2$ or to subcritical Sobolev space $H^{s}(\R^d)$ with $s>0$. We here prove that there exists some positive constant $\varepsilon_0$ depending on $d$, such that the Cauchy problem is locally and globally well-posed for any initial data $h_0$ which is radial, supported away from origin and in the negative Sobolev space $\dot H^{-\varepsilon_0}(\R^d)$ including $L^p(\R^d)$ with certain $p<2$ as subspace. Furthermore, unconditional uniqueness, and $L^2$-estimate both as time $t\to0$ and $t\to +\infty$ were considered.