I am reading a book about the Harmonic Map flow and in the proof of the existence of solution the author states something I am not totally familiar with. Let me recall the statement of the theorem.
Let $(M, g)$ and $(N, h) \subset \mathbb R^L$ be two compact Riemannian manifold without boundaries. Then for $u_0 \in C^\infty(M, N)$ there exists a maximal $T \in (0,+\infty]$ such that \begin{cases} \partial_t u - \Delta_gu = A(u)(\nabla u, \nabla u) & \text{in } M \times (0,+\infty)\\ u|_{t = 0} = u_0 \end{cases} admits a unique, smooth solution $u \in C^\infty(M\times [0,T), N)$. Moreover, if $T < \infty$, then $$\lim_{t \to T} \|\nabla u(\cdot, t)\|_{L^\infty} = \infty.$$
Where $A(u)(\cdot, \cdot)$ is the second fundamental form.
My question is about last part of the statement, when $T< \infty$. The author just claims this fact "by parabolic regularity as this equation is a quasi-linear parabolic system" [sic]. I have never seen nor used such "parabolic regularity" argument so I was wondering if one of you could tell me more about it, or give me a reference with the proof ?