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}.