About an estimate of theorem 3 in Chapter 12 of Evans' book

Question

This is the proof of theorem 3 in Chapter 12 of Evans' book as the following picture. I really don't understand why $|F(Du,u_t,u)|\le C(|Du|+|u_t|+|u|)$, because he didn't give us any restriction on $f$. why just according to $f(0,0,0)=0$ and $u$ is smooth we could derive this estimate? Thanks!

Answer 1

Consider the following statement, which does not involve any PDE: this is multivariable calculus only.

If $f:\mathbb R^n\to \mathbb R$ is $C^1$ smooth and $f(0,0,0)=0$, then for every $R>0$ there exists $C>0$ such that $|f(z)|\le C|z|$ whenever $|z|\le R$.

Proof: by the mean-value inequality we have $|f(z)-f(0)|\le |z|\sup_{|\xi|<R} |Df(\xi)|$, which yields the claim. $\quad \Box$

In your situation the argument of $f$ is $z=(Du,u_t,u)$. This triple is bounded by the smoothness of $u$; see the line where Evans refers to its $L^\infty$ norm. This gives $R$, and the aforementioned fact gives $C$.