2.3. HEAT EQUATION 61
Since |v|, |˜| v ≤ A for some constant A, Theorem 7 implies v ≡ ˜; v that is,
(36) v(x, t) =
t
0 Rn
Φ(x − y, t −
s)˜(y,
f s) dyds.
Now suppose (x, t) ∈ C . As ζ ≡ 0 off the cylinder C, (34) and (36) imply
u(x, t) =
C
Φ(x − y, t − s)[(ζs(y, s) − Δζ(y, s))u(y, s)
− 2Dζ(y, s) · Du(y, s)] dyds.
Note in this equation that the expression in the square brackets vanishes in
some region near the singularity of Φ. Integrate the last term by parts:
(37)
u(x, t) =
C
[Φ(x − y, t − s)(ζs(y, s) + Δζ(y, s))
+ 2DyΦ(x − y, t − s) · Dζ(y, s)]u(y, s) dyds.
We have proved this formula assuming u ∈
C∞.
If u satisfies only the
hypotheses of the theorem, we derive (37) with uε = ηε ∗ u replacing u, ηε
being the standard mollifier in the variables x and t, and let ε → 0.
3. Formula (37) has the form
(38) u(x, t) =
C
K(x, t, y, s)u(y, s) dyds ((x, t) ∈ C ),
where
K(x, t, y, s) = 0 for all points (y, s) ∈ C ,
since ζ ≡ 1 on C . Note also K is smooth on C − C . In view of expression
(38), we see u is
C∞
within C = C(x0,t0;
1
2
r).
c. Local estimates for solutions of the heat equation. Let us now
record some estimates on the derivatives of solutions to the heat equa-
tion, paying attention to the differences between derivatives with respect
to xi (i = 1,...,n) and with respect to t.
THEOREM 9 (Estimates on derivatives). There exists for each pair of
integers k, l = 0, 1,... a constant Ck,l such that
max
C(x,t; r/2)
|DxDtu|
k l
≤
Ckl
rk+2l+n+2
u
L1(C(x,t;r))
for all cylinders C(x, t; r/2) ⊂ C(x, t; r) ⊂ UT and all solutions u of the heat
equation in UT .
