Blowup for $u_t = \Delta u + |\nabla u|^2 u$ from $\mathbb {R}^n$ into $\mathbb {R}^m$

Abstract:

In this note we consider the global regularity of smooth solutions $u=(u^1,\dots , u^m)$ to the vector-valued Cauchy problem \[ u_t = \Delta u + |\nabla u|^2 u \quad \text {in } \mathbb {R}^n \times [0,\infty ), \quad u(x,0) = u_0(x) \quad \text {in } \mathbb {R}^n. \] We show that if $n,m \geq 3$, the gradient-blowup phenomenon occurs in finite time for suitably chosen $u_0$ vanishing at infinity. We also present a simple example of the $L^\infty$-blowup solutions for $|u_0| \equiv 1+\epsilon$ for any $\epsilon >0$, if $m \geq 2$.