1547671870-The_Ricci_Flow__Chow

102 4. MAXIMUM PRINCIPLES

Then if every solution of the ODE

d

dt a = F (a)

a (0) E JCx (0)

defined in each fiber Ex remains in JCx (t), the solution a (t) of the PDE

remains in JC (t).

The final generalization we will state is an enhanced version of the result

above. It is motivated by applications in which the ODE solution might

escape from a certain subset Ac JC, but where we may be able to assume
that the solution of the PDE avoids A. Accordingly, we call A (t) C JC (t)
the avoidance set.
THEOREM 4.10 (tensor maximum principle, fourth version: ODE controls
PDE outside avoidance sets). Adopt the assumptions above, where JC ( t) is a
closed subset of E for all t E [O, T]. Let a (t): 0 :S: t :S: T be a solution of the
nonlinear PDE
8 A
at a = Lia + F (a)
such that a (0) E JC (0). Assume further that:

• the space-time tracks UtE[O,T] (JC (t) x {t}) and UtE[O,T] (A (t) x {t})
  are closed subsets of E x [O, T];

• JC (t) is invariant under parallel translation by V (t) for all t E

[O, T);

• JCx (t) ~ JC (t) n 7r - l (x) is a closed convex subset of Ex for all
  x E Mn and t E [O, T]; and


• a (t) avoids A (t), meaning that a (t) ~A (t) for any t E [O, T].

Then if every solution of the ODE

d

dta = F (a)

a (0) E JCx (0) \Ax (0)

defined in each fiber Ex either remains in JCx ( t) for all t E [O, T] or else

enters A (to) at some time to E (0, T], the solution a (t) of the PDE remains

in JC (t).

In the successor to this volume, we will present detailed proofs of all the

results stated in this section.

4. Strong maximum principles
   It is well known that any nonnegative solution u of the heat equation
   Ut =Liu
   has u > 0 for all t > 0 unless u = 0. (See [110].) In particular, if one

knows at t = 0 that u 2:: 0 everywhere and that u > 0 at one point, one can