1547671870-The_Ricci_Flow__Chow

l. SCALAR EQUATIONS 95

1.2. The heat equation with a linear reaction term. More gen-
erally, one may add in a reaction term. We first consider the case where the
reaction term is linear. In particular, if g ( t) is a 1-parameter family of met-
rics, X (t) is a 1-parameter family of vector fields, and f3: Mn x [O, T) --+IR
is a given function, we say u is a supersolution to the linear heat equation

av

at = b..g(t)V + (X, Vv) + (Jv

at any points and times where

OU

(4.4) at 2 b..g(t)U + (X, Vu)+ (Ju.

PROPOSITION 4.3 (scalar maximum principle, third version: linear re-

action terms preserve lower bounds). Let u: Mn x [O, T) --+IR be a C^2 su-

persolution to (4.4) on a closed manifold. Suppose that for each TE [O, T),

there exists a constant C 7 < oo such that f3 (x, t) :::::; C 7 for all x E Mn and

t E [0,T]. Ifu(x,O) 2 0 for all x E Mn, then u(x,t) 2 0 for all x E Mn

and t E [O, T).

PROOF. Given TE (0, T), define

J (x , t) ~ e-Crtu (x , t),

where C 7 is as in the hypothesis. One computes that

f)J

8t 2 b.. 9 (t)J + (X, VJ)+ (/3 - Cr) J.

Since f3 - C 7 :::::; 0 on Mn x [O, T], one has

f)J

8t (x, t) 2 (b..g(t)J) (x, t) + (X, VJ) (x, t)

for all (x, t) E Mn x [O, T] such that J (x, t) :::::; 0. By Theorem 4.2, one

concludes that J 2 0 on Mn x [O, T). Hence u 2 0 on Mn x [O, T). Since

T E (0, T) was arbitrary, the proposition follows. D

1.3. The heat equation with a nonlinear reaction term. Now

we treat the case where the reaction term is nonlinear. In particular, we

consider the semilinear heat equation

av

(4.5) at = b..g(t)V + (X, Vv) + F (v)

where g (t) is a 1-parameter family of metrics, X (t) is a 1-parameter family

of vector fields, and F : IR --+ IR is a locally Lipschitz function. We say u is

a supersolution of (4.5) if

OU

at 2 b..g(t)U + (X, Vu)+ F (u)

and a subsolution if

OU

at :S b..g(t)U + (X, Vu)+ F (u).