Slowly Oscillating Solution of the Cubic Heat Equation

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.

References in corpus (2)

• Ill-posedness of the Navier-Stokes equations in a critical space in 3D
• The Fujita phenomenon in exterior domains under the Robin boundary conditions