Global existences and asymptotic behavior for semilinear heat equation

TL;DR

This paper establishes conditions for global existence and analyzes the long-term decay of solutions to the critical semilinear heat equation, extending known results to broader initial data spaces.

Contribution

It proves local and global well-posedness for initial data in negative Sobolev spaces, including some data outside traditional Lebesgue spaces, and characterizes the asymptotic decay of solutions.

Findings

Existence of solutions for initial data in negative Sobolev spaces.

Global solutions for radial, support-away-from-origin initial data.

Asymptotic decay estimates for solutions as time approaches infinity.

Abstract

In this paper, we consider the global Cauchy 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 $s>0$. {\it First,} we prove that there exists some positive constant $\gamma_0$ 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 the origin and in the negative Sobolev space $\dot H^{-\gamma_0}(\R^d)$. In particular, it leads to local and global existences of the solutions to Cauchy problem considered above for the initial data in a proper subspace of $L^p(\R^d)$ with some $p<2$. {\it Secondly,} the sharp asymptotic behavior of the solutions ( i.e. $L^2$-decay estimates ) as $t\to +\infty$ are obtained with arbitrary large initial data $h_0\in \dot H^{-\gamma_0}(\R^d)$ in the defocusing case and in the focusing case with suitably small initial data $h_0$.