60 2. FOUR IMPORTANT LINEAR PDE
to denote the closed circular cylinder of radius r, height
r2,
and top center
point (x, t).
Fix (x0,t0) ∈ UT and choose r 0 so small that C := C(x0,t0; r) ⊂ UT .
Define also the smaller cylinders C := C(x0,t0;
3
4
r), C := C(x0,t0;
1
2
r),
which have the same top center point (x0,t0).
Choose a smooth cutoff function ζ = ζ(x, t) such that
0 ≤ ζ ≤ 1, ζ ≡ 1 on C ,
ζ ≡ 0 near the parabolic boundary of C.
Extend ζ ≡ 0 in
(Rn
× [0,t0]) − C.
2. Assume temporarily that u ∈
C∞(UT
) and set
v(x, t) := ζ(x, t)u(x, t) (x ∈
Rn,
0 ≤ t ≤ t0).
Then
vt = ζut + ζtu, Δv = ζΔu + 2Dζ · Du + uΔζ.
Consequently
(33) v = 0 on
Rn
× {t = 0},
and
(34) vt − Δv = ζtu − 2Dζ · Du − uΔζ =:
˜
f
in
Rn
× (0,t0). Now set
˜(x, v t) :=
t
0
Rn
Φ(x − y, t −
s)˜(y,
f s) dyds.
According to Theorem 2
(35)
˜t v − Δ˜ v =
˜
f in
Rn
× (0,t0)
˜ v = 0 on
Rn
× {t = 0}.
