Partial Differential Equations

∫
I ≤ ε Φ(x − y, t)dy = ε.
R n
1.3 The Heat Equation 21
To estimate J, we note that for x ∈ B(x 0 , δ 2 ) and y ∉ B(x0 , δ), we have
|y − x 0 | ≤ |y − x| + δ 2 ≤ |y − x| + 1 2 |y − x0 |,
i.e. 1 2 |y − x0 | ≤ |x − y|. If ‖g‖ ∞ = sup x∈R n |g(x)|, then we find
δ
δ/2
x 0
y
x
Fig. 1.9.
J ≤ 2‖g‖ ∞
∫
≤ C′
t n 2
∫ ∞
δ
R n \B(x 0 ;δ)
Φ(x − y, t) dy = C t n 2
e − r2
16t r n−1 dr,
∫
R n \B(x 0 ;δ)
e − |x−y|2
4t dy ≤ C t n 2
∫
R n \B(x 0 ;δ)
e − |x0 −y| 2
16t
dy
and the latter integral converges to 0 for t ↘ 0, since for suitable constants c, d > 0, we have (Exercise!)
e − r2
16t
r n−1 ≤ ce − r2
t n dt ∀r ≥ δ.
2
2·10 10 0.002
1·10 10
0.0018
0
0.0016
50 100 0.0014
150
t
200 0.0012
r
250 300 0.001
8·10 10
6·10 10
0.002
4·10 10
2·10 10
0.0018
0
0.0016
50 100 0.0014
150
t
200 0.0012
r
250 300 0.001
Fig. 1.10. The functions e − r2
16t r 3 t − 3 2
and 10 11 e − r2
17t for r ∈ [50, 300] and t ∈ [10 −3 , 2 · 10 −3 ]
Together, we obtain |u(x, t) − g(x 0 )| < 2ε for sufficiently small t, and this is precisely the claim.
Remark 1.3.5. If g ∈ C(R n ) is a bounded, nonnegative and nonvanishing function, then