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

In this note we consider the global regularity of smooth solutions $u=(u^1,\dots , u^m)$ to the vector-valued Cauchy problem

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 $\vert u_0\vert \equiv 1+\epsilon$for any $\epsilon >0$, if $m \geq 2$.