Blowup for ut=Δu+|delu|2u from Rn into Rm

In this note we consider the global regularity of smooth solutions u = (u(1),..., u(m)) to the vector-valued Cauchy problem u(t) = Delta u + |del u|(2)u in R-n x [0,infinity), u(x, 0) = u(0)(x) in R-n. We show that if n, m >= 3, the gradient-blowup phenomenon occurs in finite time for suitably chosen u0 vanishing at infinity. We also present a simple example of the L-infinity-blowup solutions for |u(0)| = 1 + epsilon for any epsilon > 0, if m >= 2.