Slowly Oscillating Solution of the Cubic Heat Equation

TL;DR

This paper investigates the nonlinear heat equation with cubic nonlinearity, showing that small initial data in certain Besov spaces can lead to finite-time blowup of solutions, even after very short times.

Contribution

It extends previous results by demonstrating finite-time blowup for small initial data in specific Besov spaces, highlighting the delicate balance between initial data size and solution behavior.

Findings

Small initial data in $ ext{dot}B^{-2/3, ext{infty}}_9$ can cause finite-time blowup.

Blowup can occur arbitrarily soon after initial time.

Solutions can exhibit explosive behavior despite small initial norms.

Abstract

In this paper, we are considering the Cauchy problem of the nonlinear heat equation $u\_t -\Delta u= u^{3 },\ u(0,x)=u\_0$. After extending Y. Meyer's result establishing the existence of global solutions, under a smallness condition of the initial data in the homogeneous Besov spaces $\dot{B}\_{p}^{-\sigma, \infty}(\mathbb{R}^{3})$, where $3 \textless{} p \textless{} 9$ and $\sigma=1-3/p$, we prove that initial data $u\_0\in \mathcal{S}(\mathbb{R}^{3})$, arbitrarily small in ${\dot B^{-2/3,\infty}\_{9}}(\mathbb{R}^{3})$, can produce solutions that explode in finite time. In addition, the blowup may occur after an arbitrarily short time.