Jost J. Partial Differential Equations

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4.2 The Fundamental Solution of the Heat Equation 97

Δw(x, t) −

∂

∂t

w(x, t)=φ(x, t) in Ω × (0, ∞),

w(x, t)=γ(x, t) in ∂Ω × (0, ∞). (4.2.27)

Then

lim

t→∞

w(x, t)=0 uniformly in x ∈ Ω. (4.2.28)

Proof: We choose R>0 such that

<R for all x =(x

,...,x

) ∈ Ω, (4.2.29)

and consider

k(x):=e

− e

. (4.2.30)

Then

Δk = −e

With κ := inf

x∈Ω

,wethushave

Δk ≤−κ. (4.2.31)

We consider, with constants η,c

,τ to be determined, and with

:= inf

x∈Ω

k(x),κ

:= sup

x∈Ω

k(x),

the expression

m(x, t):=η

k(x)

+ η

k(x)

+ c

k(x)

−

(t−τ)

(4.2.32)

in Ω ×[τ,∞).

Then

Δm(x, t) −

∂

∂t

m(x, t)

< −η − η

− c

−

(t−τ)

+ c

−

(t−τ)

< −η. (4.2.33)

Furthermore,

m(x, τ ) >c

for x ∈ Ω, (4.2.34)

m(x, t) >η for (x, t) ∈ ∂Ω × [τ,∞). (4.2.35)

By our assumptions (4.2.25), (4.2.26), for every η, there exists some τ = τ(η)

with

98 4. Existence Techniques II: Parabolic Methods. The Heat Equation

|φ(x, t)| <η for x ∈ Ω, t ≥ τ, (4.2.36)

|γ(x, t)| <η for x ∈ ∂Ω, t ≥ τ. (4.2.37)

In (4.2.32) we now put

τ = τ(η),c

=sup

x∈Ω

|w(x, τ )|.

Then

m(x, τ ) ± w(x, τ ) ≥ 0forx ∈ Ω by (4.2.34),

m(x, t) ± w(x, t) ≥ 0forx ∈ ∂Ω, t ≥ τ,

by (4.2.35), (4.2.37), (4.2.27);



Δ −

∂

∂t



(m(x, t) ± w(x, t)) ≤ 0forx ∈ Ω, t ≥ τ,

by (4.2.33), (4.2.36), (4.2.27).

It follows from Theorem 4.1.1 (observe that it is irrelevant that our functions

are deﬁned only on Ω × [τ,∞) instead of Ω × [0, ∞), and initial values are

given on Ω ×{τ})that

|w(x, t)|≤m(x, t)forx ∈ Ω, t > τ,

≤ η





+ c

−

(t−τ)

and this becomes smaller than any given ε>0ifη>0 from (4.2.36), (4.2.37)

is suﬃciently small and t>τ(η) is suﬃciently large. 

4.3 The Initial Boundary Value Problem

for the Heat Equation

In this section, we wish to study the initial boundary value problem for the

inhomogeneous heat equation

(x, t) − Δu(x, t)=ϕ(x, t)forx ∈ Ω, t > 0,

u(x, t)=g(x, t)forx ∈ ∂Ω,t > 0,

u(x, 0) = f(x)forx ∈ Ω,

(4.3.1)

with given (continuous and smooth) functions ϕ, g, f. We shall need some

preparations.

Lemma 4.3.1: Let Ω be a bounded domain of class C

in R

. Then for every

α<

+1, T>0 there exists a constant c = c(α, T, d, Ω) such that for all

,x∈ ∂Ω, 0 <t≤ T , letting ν denote the exterior normal of ∂Ω, we have



∂K

∂ν

(x, x

,t)



≤ ct

−α

|x − x

−d+2α

4.3 The Initial Boundary Value Problem for the Heat Equation 99

Proof:

∂

∂ν

K(x, x

,t)=

(4πt)

∂

∂ν

−

|x−x

= −

(4πt)

(x − x

) · ν

−

|x−x

As we are assuming that the boundary of Ω is a manifold of class C

,and

since x, x

∈ ∂Ω,andν

is normal to ∂Ω,wehave

|(x − x

) · ν

|≤c

|x − x

with a constant c

depending on the geometry of ∂Ω.Thus



∂

∂ν

K(x, x

,t)



≤ c

−

−1

|x − x

−

|x−x

(4.3.2)

with some constant c

. With a parameter β>0, we now consider the function

ψ(s):=s

−s

for s>0. (4.3.3)

Inserting s =

|x−x

, β =

+1− α, we obtain from (4.3.3)

−

|x−x

≤ c

|x − x

−d−2+2α

+1−α

, (4.3.4)

with c

depending on β, i.e., on d and α. Inserting (4.3.4) into (4.3.2) yields

the assertion. 

Lemma 4.3.2: Let Ω ⊂ R

be a bounded domain of class C

with exterior

normal ν,andletγ ∈ C

(∂Ω × [0,T]) (T>0). We put

v(x, t):=−



∂Ω

∂K

∂ν

(x, y, τ)γ(y,t −τ)do(y)dτ. (4.3.5)

We then have

v ∈ C

∞

(Ω ×[0,T]),

v(x, 0) = 0 for all x ∈ Ω, (4.3.6)

and for all x

∈ ∂Ω, 0 <t≤ T ,

lim

x→x

v(x, t)=

γ(x

,t)

−



∂Ω

∂K

∂ν

,y,τ)γ(y, t −τ)do(y)dτ. (4.3.7)

Proof: First of all, Lemma 4.3.1, with α =

, implies that the integral in

(4.3.5) indeed exists. The C

∞

-regularity of v with respect to x then follows

from the corresponding regularity of the kernel K by the change of variables

σ = t − τ . Equation (4.3.6) is obvious as well. It remains to verify the jump

relation (4.3.7). For that purpose, it obviously suﬃces to investigate

100 4. Existence Techniques II: Parabolic Methods. The Heat Equation

−



∂Ω∩B(x

,δ)

∂K

∂ν

(x, y, τ)γ(y,t −τ)do(y)dτ (4.3.8)

for arbitrarily small τ

> 0, δ>0. In particular, we may assume that δ

and

τ are chosen such that for any given ε>0, we have for y ∈ ∂Ω, |y − x

| <δ,

and 0 ≤ τ<τ

|γ(x

,t) −γ(y, t − τ )| <ε.

Thus, we shall have an error of magnitude controlled by ε if in place of (4.3.8),

we evaluate the integral

−



∂Ω∩B(x

,δ)

∂K

∂ν

(x, y, τ)γ(x

,t)do(y)dτ. (4.3.9)

Extracting the factor γ(x

,t) it remains to show that

− lim

x→x



∂Ω∩B(x

,δ)

∂K

∂ν

(x, y, τ)do(y)dτ =

+ O(δ). (4.3.10)

Also, we observe that since γ is continuous, it suﬃces to show that (4.3.10)

holds uniformly in x

if x approaches ∂Ω in the direction normal to ∂Ω.In

other words, letting ν(x

) denote the exterior normal vector of ∂Ω at x

,we

may assume

x = x

− μν(x

In that case, μ

= |x −x

, and since ∂Ω is of class C

,fory ∈ ∂Ω,

|x − y|

= |y − x

+ μ

+ O



|y − x

|x − x



The term O



|y − x

|x − x



here is a higher-order term that does not

inﬂuence the validity of our subsequent limit processes, and so we shall omit

it in the sequel for the sake of simplicity. Likewise, for y ∈ ∂Ω,

(x − y) · ν

=(x − x

) · ν

+(x

− y) · ν

= −μ + O



− y|



and the term O(|x

− y|

) may be neglected again.

Thus we approximate

∂K

∂ν

(x, y, τ)=

(4πτ)

(x − y) · ν

2τ

−

|x−y|

4τ

(4πτ)

(−μ)

2τ

−

−y|

4τ

−

4τ

4.3 The Initial Boundary Value Problem for the Heat Equation 101

This means that we need to estimate the expression



∂Ω∩B(x

,δ)

2(4π)

−

−y|

4τ

−

4τ

do(y)dτ.

We introduce polar coordinates with center x

and put σ = |x

− y|.Wethen

obtain, again up to a higher-order error term,

μVol(S

d−2

)

2(4π)



−

4τ



−

4τ

d−2

dr dτ,

where S

d−2

is the unit sphere in R

d−1

μVol(S

d−2

)

4π



−

4τ



2τ

−s

d−2

ds dτ

Vol(S

d−2

)

2π



∞

4τ

−σ



δσ

−s

d−2

ds dσ.

In this integral we may let μ tend to 0 and obtain as limit

Vol(S

d−2

)

2π



∞

−σ



∞

−s

d−2

ds dσ =

. (4.3.11)

By our preceding considerations, this implies (4.3.10).

Equation (4.3.11) is shown with the help of the gamma function

Γ (x)=



∞

−t

x−1

dt for x>0.

We have

Γ (x +1)=xΓ(x) for all x>0,

and because of Γ (1) = 1, then

Γ (n +1)=n!forn ∈ N.

Moreover,



∞

−s

ds =



n +1



for all n ∈ N.

In particular,







∞

−s

ds =

√

102 4. Existence Techniques II: Parabolic Methods. The Heat Equation

and



−|x|

dx = Vol(S

d−1

)



∞

−r

d−1

dr =

Vol(S

d−1

)Γ





;

hence

Vol(S

d−1

2π





With these formulae, the integral (4.3.11) becomes

2π

d−1



d−1



2π







d − 1





In an analogous manner, one proves the following lemma:

Lemma 4.3.3: Under the assumptions of Lemma 4.3.2, for

w(x, t):=



∂Ω

K(x, y, τ)γ(y,t − τ) do(y) dτ (4.3.12)

(x ∈ Ω, 0 ≤ t ≤ T ), we have

w ∈ C

∞

(Ω ×[0,T]),

w(x, 0) = 0 for x ∈ Ω. (4.3.13)

The function w extends continuously to

Ω ×[0,T],andforx

∈ ∂Ω we have

lim

x→x

∇

w(x, t) · ν(x

γ(x

,t)



∂Ω

∂K

∂ν

,y,τ)γ(y, t −τ) do(y) dτ.

(4.3.14)



We now want to try ﬁrst to ﬁnd a solution of

Δu −

∂

∂t

u =0 inΩ × (0, ∞),

u(x, 0) = 0 for x ∈ Ω,

u(x, t)=g(x, t)forx ∈ ∂Ω, t > 0, (4.3.15)

by Lemma 4.3.2.

We try

u(x, t)=−



∂Ω

∂K

∂ν

(x, y, t − τ)γ(y,τ) do(y) dτ, (4.3.16)

4.3 The Initial Boundary Value Problem for the Heat Equation 103

with a function γ(x, t) yet to be determined. As a consequence of (4.3.7),

(4.3.15), γ has to satisfy, for x

∈ ∂Ω,

g(x

,t)=

γ(x

,t) −



∂Ω

∂K

∂ν

,y,t− τ)γ(y, τ) do(y) dτ,

i.e.,

γ(x

,t)=2g(x

,t)+2



∂Ω

∂K

∂ν

,y,t− τ)γ(y, τ) do(y) dτ. (4.3.17)

This is a ﬁxed-point equation for γ, and one may attempt to solve it by

iteration; i.e., for x

∈ ∂Ω,

,t)=2g(x

,t),

,t)=2g(x

,t)+2



∂Ω

∂K

∂ν

,y,t− τ)γ

n−1

(y, τ) do(y)dτ

for n ∈ N. Recursively, we obtain

,t)=2g(x

,t)+2



∂Ω



ν=1

,y,t− τ)g(y, τ) do(y)dτ (4.3.18)

with

,y,t)=2

∂K

∂ν

,y,t),

ν+1

,y,t)=2



∂Ω

,z,t−τ)

∂K

∂ν

(z,y, τ) do(z) dτ.

In order to show that this iteration indeed yields a solution, we have to

verify that the series

S(x

,y,t)=

∞



ν=1

,y,t)

converges.

Choosing once more α =

in Lemma 4.3.1, we obtain

,y,t)|≤ct

−3/4

− y|

−(d−1)+

Iteratively, we get

,y,t)|≤c

−1+

− y|

−(d−1)+

We now choose n = max(4, 2(d − 1)) so that both exponents are positive. If

now

104 4. Existence Techniques II: Parabolic Methods. The Heat Equation

,y,t)|≤β

for some constant β

and some α ≥ 0,

then

m+1

,y,t)|≤cβ



(t − τ)

−3/4

dτ,

where the constant c comes from Lemma 4.3.1 and

:= sup

y∈∂Ω



∂Ω

|z − y|

−(d−1)+

do(z).

Furthermore,



(t − τ)

−3/4

dτ =

Γ (1 + α)Γ







+ α



α+1/4

where on the right-hand side we have the gamma function introduced above.

Thus

n+ν

,y,t)|≤β

(cβ

)

α+ν/4

μ=1



α +

+ μ/4







Γ (α +1+μ/4)

Since the gamma function grows factorially as a function of its arguments,

this implies that

∞



ν=1

,y,t)

converges absolutely and uniformly on ∂Ω ×∂Ω ×[0,T] for every T>0. We

thus have the following result:

Theorem 4.3.1: The initial boundary value problem for the heat equation

on a bounded domain Ω ⊂ R

of class C

, namely,

Δu(x, t) −

∂

∂t

u(x, t)=0 in Ω ×(0, ∞),

u(x, 0) = 0 in Ω,

u(x, t)=g(x, t) for x ∈ ∂Ω, t > 0,

with given continuous g, admits a unique solution. That solution can be rep-

resented as

u(x, t)=−



∂Ω

Σ(x, y, t − τ)g(y, τ) do(y) dτ, (4.3.19)

where

Σ(x, y, t)=2

∂K

∂ν

(x, y, t)+2



∂Ω

∂K

∂ν

(x, z, t − τ )

∞



ν=1

(z,y, τ) do(z) dτ.

(4.3.20)

4.3 The Initial Boundary Value Problem for the Heat Equation 105

Proof: Since the series



∞

ν=1

converges,

γ(x

,t)=2g(x

,t)+2



∂Ω

∞



ν=1

,y,t− τ)g(y, τ) do(y) dτ

is a solution of (4.3.17). Inserting this into (4.3.16), we obtain (4.3.20). Here,

one should note that

−3/4

|y − x|

−(d−1)+

∞



ν=1

,y,τ),

and hence also Σ(x, y, t) converges absolutely and uniformly on ∂Ω × ∂Ω ×

[0,T] for every T>0. Thus, we may diﬀerentiate term by term under the

integral and show that u solves the heat equation. The boundary values are

assumed by construction, and it is clear that u vanishes at t = 0. Uniqueness

follows from Theorem 4.1.1. 

Deﬁnition 4.3.1: Let Ω ⊂ R

be a domain. A function q(x, y, t) that is

deﬁned for x, y ∈

Ω, t>0, is called the heat kernel of Ω if

(i)



−

∂

∂t



q(x, y, t)=0 for x, y ∈ Ω, t > 0, (4.3.21)

(ii)

q(x, y, t)=0 for x ∈ ∂Ω, (4.3.22)

(iii) and for all continuous f : Ω → R

lim

t→0



q(x, y, t)f(x)dx = f (y) for all y ∈ Ω. (4.3.23)

Corollary 4.3.1: Any bounded domain Ω ⊂ R

of class C

has a heat ker-

nel, and this heat kernel is of class C

Ω with respect to the spatial vari-

ables y. The heat kernel is positive in Ω, for all t>0.

Proof: For each y ∈ Ω, by Theorem 4.3.1 we solve the boundary value prob-

lem for the heat equation with initial values 0 and

g(x, t)=−K(x, y, t).

The solution is called μ(x, y, t), and we put

q(x, y, t):=K(x, y, t)+μ(x, y, t). (4.3.24)

Obviously, q(x, y, t) satisﬁes (i) und (ii), and since

lim

t→0

μ(x, y, t)=0,

106 4. Existence Techniques II: Parabolic Methods. The Heat Equation

and K(x, y, t) satisﬁes (iii), then so does q(x, y, t).

Lemma 4.3.3 implies that q canbeextendedto

Ω as a continuously dif-

ferentiable function of the spatial variables.

That q(x, y, t) > 0 for all x, y ∈ Ω, t > 0 follows from the strong maximum

principle (Theorem 4.1.3). Namely,

q(x, y, t)=0 forx ∈ ∂Ω,

q(x, y, t)=0 forx, y, ∈ Ω,x = y,

while (iii) implies

q(x, y, t) > 0if|x − y| and t>0 are suﬃciently small.

Thus, q ≥ 0andq = 0, and so by Theorem 4.1.3,

q>0inΩ ×Ω × (0, ∞).



Lemma 4.3.4 (Duhamel principle): For all functions u, v on Ω × [0,T]

with the appropriate regularity conditions, we have





v(x, t)(Δu(x, T − t)+u

(x, T − t))

− u(x, T − t)(Δv(x, t) − v

(x, t))



dx dt



∂Ω



∂u

∂ν

(y, T −t)v(y, t) −

∂v

∂ν

(y, t)u(y,T − t)



do(y) dt



{u(x, 0)v(x, T ) − u(x, T )v(x, 0)} dx. (4.3.25)

Proof: Same as the proof of (4.1.12). 

Corollary 4.3.2: If the heat kernel q(z, w,T) of Ω is of class C

Ω with

respect to the spatial variables, then it is symmetric with respect to z and w,

i.e.,

q(z, w, T)=q(w,z, T) for all z,w ∈ Ω, T > 0. (4.3.26)

Proof: In (4.3.25), we put u(x, t)=q(x, z, t), v(x, t)=q(x, w, t). The double

integrals vanish by properties (i) and (ii) of Deﬁnition 4.3.1. Property (iii) of

Deﬁnition 4.3.1 then yields v(z, T)=u(w, T), which is the asserted symmetry.



Theorem 4.3.2: Let Ω ⊂ R

be a bounded domain of class C

with heat

kernel q(x, y, t) according to Corollary 4.3.1, and let

ϕ ∈ C

(

Ω ×[0, ∞)),g∈ C

(∂Ω × (0, ∞)),f∈ C

(Ω).