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}.
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