1. WEAK MAXIMUM PRINCIPLES FOR SCALARS AND 2-TENSORS 3

associated to g (t). Given a C°° function u : M x [0, T) — • M, we define the

semilinear second-order parabolic operator:

Ov

(io.i)

Lu^Yt~ As(i)U

~

x ( t )

'

V u )

~

F (n'*}'

where X (t) is a time-dependent vector field and the function F : Ex [0, T) —

K. is locally Lipschitz in the M factor and continuous in the [0, T) factor.

The essence of the scalar maximum principle is contained in the following

comparison result.

THEOREM

10.1 (Comparison principle). Suppose that u and v satisfy

Lu Lv on M x [0, T) and u (x, 0) v (x, 0) for all x G M. Then

u (x, t) v (or, t) onMx [0, T).

This theorem can be proved in the same manner as for Theorem 4.4

on p. 96 of Volume One using the second derivative test. As a special case

we may take v = v (t) to be a function depending only on time and may

consider the ODE Lv — ^ — F (v,i) = 0, which is no longer a PDE since

the Laplacian term and the gradient term vanish. Hence, as a consequence,

we have the following two scalar weak maximum principles. We have the

following result for subsolutions.

THEOREM

10.2 (Scalar maximum principle (SMP)). Suppose that u sat-

isfies Lu 0; where L is defined by (10.1), and suppose u(x,0) c for all

x G M and some constant c G R. Let (f)(t) be the solution to the correspond-

ing ordinary differential equation:

(10.2) ^ = F ( 0 , t ) ,

(10.3) 0(0) = c.

Then

(10.4) u{x,t) cj)(t)

for all x G Ai and t G [0,T); as long as the solution to the ODE exists.

We have the analogous result for supersolutions where we replace by

in the statement of the above theorem. For a solution we obtain both

upper and lower bounds.

THEOREM

10.3 (SMP—upper and lower bounds). Suppose that u satis-

fies Lu = 0 for L as in (10.1) and suppose c\ u (x, 0) C 2 for all x G M,

where c\,C2 G M. Let (fi(t), i — 1,2; be the solutions to the associated ODE;

K0) = ci.

Then

Mt)u(x,t)42(t)

for all x G M and t G [0, T), as long as the solutions to the ODE exist.