Introduction to Partial Differential Equations
Exercise Sheet 11

Ross Ogilvie      11th November, 2024
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33.
The distribution of heat

Consider the fundamental solution of the heat equation Φ(x,t) given in Definition 4.5.

(a)
Show that this extends to a smooth function on n × {(0,0)}. (2 points)
(b)
Verify that this obeys the heat equation on n × {(0,0)}. (2 points)

We want to show that φH(φ) = n×Φ(x,t)φ(x,t)𝑑𝑥𝑑𝑡 is a distribution. Clearly it is linear. Fix a set K n × and let φ C0(K).

(c)
Why must there be a constant T > 0 with
H(φ) =0T nΦ(x,t)φ(x,t)𝑑𝑥𝑑𝑡?

(1 point)

(d)
Conclude with the help of Lemma 4.6 and Theorem 4.7 that
|H(φ)| TφK,0.

Hence H is a continuous linear functional. (2 points)

Finally, we want to show that (in the sense of distributions) (t Δ)H = δ.

(e)
Extend Theorem 4.7 to show that
nΦ(x y,t)h(y,s)𝑑𝑦 h(x,s)

as t 0, uniformly in s. (1 point)

(f)
Hence show that
𝜀nΦ(tφ Δφ)𝑑𝑦𝑑𝑡 φ(0,0)

as 𝜀 0. (3 points)

(g)
Prove that as 𝜀 0
0𝜀nΦ(y,t)h(y,t)𝑑𝑦𝑑𝑡 0

(2 points)

Together these integrals show that

(t Δ)H(φ) = (0𝜀 +𝜀)nΦ(tφ Δφ)𝑑𝑦𝑑𝑡 = φ(0,0) = δ(φ)

for all test functions φ. Therefore (t Δ)H = δ as claimed.

34.
Heat death of the universe

First a corollary to Theorem 4.7:

(a)
Suppose that h Cb(n) L1(n) and u is defined as in Theorem 4.7. Show
sup xn|u(x,t)| 1 (4𝜋𝑡)n2hL1.

(2 points)

The above corollary shows how solutions to the heat equation on n × + with such initial conditions behave: they tend to zero as t . Physically this is because if h L1 then there is a finite amount of total heat, which over time becomes evenly spread across the space.

On open and bounded domains Ω n we can have different behaviour, due to the boundary conditions holding the temperature steady. In this question we determine the long time behaviour of solutions u to the heat equation on open and bounded sets Ω with u(x,t) = 0 on Ω × + and u(x,0) = h(x).We claim u 0 as t .

(b)
Let lm be the function from Theorem 4.7 that solves heat equation on n with lm(x,0) = 𝑚𝑘(x) for m a constant and k : n [0,1] a smooth function of compact support such that k|Ω 1. Why must k exist? Why does lm 0 as t ? What boundary conditions on Ω does it obey? (3 points)
(c)
Use the monotonicity property to show that u tends to zero. (2 points)

Hint. Consider a = sup xΩ|u(x,0)|.