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4.3 The Initial Boundary Value Problem for the Heat Equation 107

Then the initial boundary value problem

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

u(x, t)=g(x, t) for x ∈ ∂Ω, t > 0,

u(x, 0) = f(x) for x ∈ Ω, (4.3.27)

admits a unique solution that is continuous on

Ω × [0, ∞) \∂Ω ×{0} and is

represented by the formula

u(x, t)=



q(x, y, t − τ)ϕ(y, τ)dy dτ



q(x, y, t)f(y)dy −



∂Ω

∂q

∂ν

(x, y, t − τ)g(y, τ)do(y)dτ. (4.3.28)

Proof: Uniqueness follows from the maximum principle. We split the exis-

tence problem into two subproblems:

We solve

(x, t) − Δv(x, t)=0 forx ∈ Ω, t > 0,

v(x, t)=g(x, t)forx ∈ ∂Ω,t > 0,

v(x, 0) = f (x)forx ∈ Ω,

(4.3.29)

i.e., the homogeneous equation with the prescribed initial and boundary con-

ditions, and

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

w(x, t)=0 forx ∈ ∂Ω,t > 0,

w(x, 0) = 0 for x ∈ Ω,

(4.3.30)

i.e., the inhomogeneous equation with vanishing initial and boundary values.

The solution of (4.3.27) is then given by

u = v + w.

We ﬁrst address (4.3.29), and we claim that the solution v can be represented

v(x, t)=



q(x, y, t)f(y)dy −



∂Ω

∂q

∂ν

(x, y, t − τ)g(y, τ)do(y)dτ.

The facts that v solves the heat equation and the initial condition v(x, 0) =

f(x) follow from the corresponding properties of q.Moreover,q(x, y, t)=

K(x, y, t)+μ(x, y, t)withμ(x, y, t) coming from the proof of Corollary 4.3.1.

By Theorem 4.3.1, this μ can be represented as

μ(x, y, t)=



∂Ω

Σ(x, z, t − τ )K(z, y,τ)do(z) dτ, (4.3.31)

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

and by Lemma 4.3.3, we have for y ∈ ∂Ω,

∂μ

∂ν

(x, y, t)=

Σ(x, y, t)



∂Ω

Σ(x, z, t − τ )

∂K

∂ν

(z,y, τ)do(z) dτ.

(4.3.32)

This means that the second integral on the right-hand side of (4.3.28) is pre-

cisely of the type (4.3.19), and thus, by the considerations of Theorem 4.3.1, v

indeed satisﬁes the boundary condition v(x, t)=g(x, t)forx ∈ ∂Ω, because

the ﬁrst integral vanishes on the boundary.

We now turn to (4.3.30). For every τ>0, we let z(x, t, τ) be the solution

(x, t; τ ) − Δz(x, t, τ)=0 forx ∈ Ω,t > τ,

z(x, t; τ)=0 forx ∈ ∂Ω,t > τ,

z(x, τ; τ)=ϕ(x, τ)forx ∈ Ω.

(4.3.33)

This is a special case of (4.3.29), which we already know how to solve, except

that the initial conditions are not prescribed at t =0,butatt = τ .This

case, however, is trivially reduced to the case of initial conditions at t =0by

replacing t by t −τ , i.e., considering ζ(x, t; τ )=z(x, t + τ ; τ ). Thus, (4.3.33)

can be solved.

We then put

w(x, t)=



z(x, t; τ)dτ. (4.3.34)

Then

(x, t)=



(x, t; τ )dτ + z(x, t; t)=



Δz(x, t; τ)dτ + ϕ(x, t)

= Δw(x, t)+ϕ(x, t)

and

w(x, t)=0 forx ∈ ∂Ω, t > 0,

w(x, 0) = 0 for x ∈ Ω.

Thus, w is a solution of (4.3.30) as required, and the proof is complete, since

the representation formula (4.3.28) follows from the one for v and the one for

w that, by (4.3.34), comes from integrating the one for z. The latter in turn

solves (4.3.33), and so by what has been proved already, is given by

z(x, t; τ)=



q(x, y, t − τ)ϕ(x, τ)dy.

Thus, inserting this into (4.3.34), we obtain

4.3 The Initial Boundary Value Problem for the Heat Equation 109

w(x, t)=



q(x, y, t − τ)ϕ(x, τ)dy dτ. (4.3.35)

This completes the proof. 

We brieﬂy interrupt our discussion of the solution of the heat equation and

record the following simple result on the heat kernel q for subsequent use:



q(x, y, t)dy ≤ 1 (4.3.36)

for all t ≥ 0. To start, we have

lim

t→0



q(x, y, t)dy =1. (4.3.37)

This follows from (4.3.23) with f ≡ 1 and the proof of Corollary 4.3.1 which

enables to replace the integration w.r.t. x in (4.3.23) by the one w.r.t. y in

(4.3.37). Next, we observe that

∂q

∂ν

(x, y, t) ≤ 0 (4.3.38)

because q is nonnegative in Ω and vanishes on the boundary ∂Ω (see (4.3.22)

and Corollary 4.3.1). We then note that the solution of Theorem 4.3.2 for

ϕ ≡ 1,g(x, t)=t, f(x)=0isgivenbyu(x, t)=t. In the representation

formula (4.3.28), using (4.3.38), this yields



q(x, y, t − τ)dy dτ ≤ t, (4.3.39)

from which (4.3.36) is derived upon a little reﬂection.

We now resume the discussion of the solution established in Theorem 4.3.2.

We did not claim continuity of our solution at the corner ∂Ω ×{0},and

in general, we cannot expect continuity there unless we assume a matching

condition between the initial and the boundary values. We do have, however,

Theorem 4.3.3: The solution of Theorem 4.3.2 is continuous on all of

Ω ×

[0, ∞) when we have the compatibility condition

g(x, 0) = f (x) for x ∈ ∂Ω. (4.3.40)

Proof: While the continuity at the corner ∂Ω ×{0} could also be established

from a reﬁnement of our previous considerations, we provide here some inde-

pendent and simpler reasoning. By the general superposition argument that

we have already employed a few times (in particular in the proof of Theorem

4.3.2), it suﬃces to establish continuity for a solution of

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

(x, t) − Δv(x, t)=0 forx ∈ Ω, t > 0,

v(x, t)=g(x, t)forx ∈ ∂Ω, t > 0,

v(x, 0) = 0 for x ∈ Ω, (4.3.41)

with a continuous g satisfying

g(x, 0) = 0 for x ∈ ∂Ω, (4.3.42)

and for a solution of

(x, t) − Δw(x, t)=0 forx ∈ Ω, t > 0,

w(x, t)=0 forx ∈ ∂Ω, t > 0,

w(x, 0) = f(x)forx ∈ Ω, (4.3.43)

with a continuous f satisfying

f(x)=0 forx ∈ ∂Ω. (4.3.44)

(We leave it to the reader to check the case of a solution of the inhomogeneous

equation u

(x, t) − Δu(x, t)=ϕ(x, t) with vanishing initial and boundary

values.)

To deal with the ﬁrst case, we consider, for τ>0,

˜v

(x, t) − Δ˜v(x, t)=0 forx ∈ Ω, t > 0,

˜v(x, t)=0 forx ∈ ∂Ω, 0 <t≤ τ,

˜v(x, t)=g(x, t − τ )forx ∈ ∂Ω, t > τ,

˜v(x, 0) = 0 for x ∈ Ω. (4.3.45)

Since, by (4.3.42), the boundary values are continuous at t = τ,bythe

boundary continuity result of Theorem 4.3.2, ˜v(x, τ) is continuous for x ∈ ∂Ω.

Also, by uniqueness, ˜v(x, t) = 0 for 0 ≤ t ≤ τ, because both the boundary and

initial values vanish there. Therefore, again by uniqueness, v(x, t)=˜v(x, t+τ )

and we conclude the continuity of v(x, 0) for x ∈ ∂Ω.

We can now turn to the second case. We consider some bounded C

domain

Ω with

Ω ⊂

Ω. We put f

(x) := max(f(x), 0) for x ∈ Ω and f(x)=0for

x ∈

Ω\Ω. Then, because of (4.3.44), f

is continuous on

Ω.Wethensolve

˜w

(x, t) − Δ ˜w(x, t)=0 forx ∈

Ω, t > 0,

˜w(x, t)=0 forx ∈ ∂

Ω, t > 0,

˜w(x, 0) = f

(x)forx ∈

Ω. (4.3.46)

By the continuity result of Theorem 4.3.2, ˜w(x, 0) is continuous for x ∈

Ω,and

therefore in particular for x ∈ ∂Ω.Sincef

(x)=0forx ∈ ∂Ω,˜w(x, t) →

0forx ∈ ∂Ω and t → 0. Since the initial values of ˜w are non-negative,

4.3 The Initial Boundary Value Problem for the Heat Equation 111

˜w(x, t) ≥ 0 for all x ∈

Ω and t ≥ 0 by the maximum principle (Theorem

4.1.1). In particular, ˜w(x, t) ≥ w(x, t)forx ∈ ∂Ω since w(x, t)=0there.

Since also ˜w(x, 0) = f

(x) ≥ f(x)=w(x, 0), the maximum principle implies

˜w(x, t) ≥ w(x, t) for all x ∈

Ω,t ≥ 0. Altogether, w(x, 0) ≤ 0forx ∈ ∂Ω.

Doing the same reasoning with f

−

(x):=min(f(x), 0), we conclude that also

w(x, 0) ≥ 0forx ∈ ∂Ω, that is, altogether, w(x, 0) = 0 for x ∈ ∂Ω.This

completes the proof. 

Remark: Theorem 4.3.2 does not claim that u is twice diﬀerentiable with re-

spect to x, and in fact, this need not be true for a ϕ that is merely continuous.

However, one may still justify the equation

(x, t) − Δu(x, t)=ϕ(x, t).

We shall return to the analogous issue in the elliptic case in Sections 10.1 and

11.1. In Section 11.1, we shall verify that u is twice continuously diﬀerentiable

with respect to x if we assume that ϕ is H¨older continuous.

Here, we shall now concentrate on the case ϕ = 0 and address the regularity

issue both in the interior of Ω and at its boundary. We recall the representa-

tion formula (4.1.14) for a solution of the heat equation on Ω,

u(x, t)=



K(x, y, t)u(y, 0) dy



∂Ω



K(x, y, t − τ)

∂u(y,τ)

∂ν

−

∂K

∂ν

(x, y, t − τ)u(y, τ)



do(y) dτ.

(4.3.47)

We put K(x, y, s)=0fors ≤ 0 and may then integrate the second integral

from 0 to ∞ instead of from 0 to t.ThenK(x, y, s)isofclassC

∞

for x, y ∈ R

s ∈ R, except at x = y, s =0.Wethushavethefollowingtheorem:

Theorem 4.3.4: Any solution u(x, t) of the heat equation in a domain Ω is

of class C

∞

with respect to x ∈ Ω, t>0.

Proof: Since we do not know whether the normal derivative

∂u

∂ν

exists on ∂Ω

and is continuous there, we cannot apply (4.3.47) directly. Instead, for given

x ∈ Ω, we consider some ball B(x, r) contained in Ω. We then apply (4.3.47)

B(x, r)inplaceofΩ.Since∂B(x, r)inΩ is contained in Ω,andu as a

solution of the heat equation is of class C

there, the normal derivative

∂u

∂ν

on ∂B(x, r) causes no problem, and the assertion is obtained. 

In particular, the heat kernel q(x, y, t) of a bounded C

-domain Ω is of

class C

∞

with respect to x, y ∈ Ω, t>0. This also follows directly from

(4.3.24), (4.3.31), (4.3.20), and the regularity properties of Σ(x, y, t) estab-

lished in Theorem 4.3.1. From these solutions it also follows that

∂q

∂ν

(x, y, t)

for y ∈ ∂Ω is of class C

∞

with respect to x ∈ Ω, t>0. Thus, one can also use

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

the representation formula (4.3.28) for deriving regularity properties. Putting

q(x, y, s)=0fors<0, we may again extend the second integral in (4.3.28)

from 0 to ∞, and we then obtain by integrating by parts, assuming that the

boundary values are diﬀerentiable with respect to t,

∂

∂t

u(x, t)=



∂

∂t

q(x, y, t)f(y)dy

−



∞



∂Ω

∂q

∂ν

(x, y, t − τ)

∂

∂τ

g(y, τ) do(y) dτ

+ lim

τ→0



∂Ω

∂g

∂ν

(x, y, t − τ)g(y, τ) do(y). (4.3.48)

Since q(x, y, t)=0forx ∈ ∂Ω, y ∈ Ω, t>0, also

∂q

∂ν

(x, y, t − τ)=0for

x, y ∈ ∂Ω,τ < t and

∂

∂t

q(x, y, t)=0 forx ∈ ∂Ω, y ∈ Ω, t > 0 (4.3.49)

(passing to the limit here is again justiﬁed by (4.3.31)). Since the second

integral in (4.3.48) has boundary values

∂

∂t

g(x, t), we thus have the following

result:

Lemma 4.3.5: Let u be a solution of the heat equation on the bounded C

domain Ω with continuous boundary values g(x, t) that are diﬀerentiable with

respect to t. Then u is also diﬀerentiable with respect to t,forx ∈ ∂Ω, t>0,

and we have

∂

∂t

u(x, t)=

∂

∂t

g(x, t) for x ∈ ∂Ω, t > 0. (4.3.50)



We are now in position to establish the connection between the heat and

Laplace equation rigorously that we had arrived at from heuristic considera-

tions in Section 4.2.

Theorem 4.3.5: Let Ω ⊂ R

be a bounded domain of class C

,andlet

f ∈ C

(Ω), g ∈ C

(∂Ω).Letu be the solution of Theorem 4.3.2 of the initial

boundary value problem

Δu(x, t) − u

(x, t)=0 for x ∈ Ω, t > 0,

u(x, 0) = f(x) for x ∈ Ω, (4.3.51)

u(x, t)=g(x) for x ∈ ∂Ω, t > 0.

Then u converges for t →∞uniformly on

Ω towards a solution of the Dirich-

let problem for the Laplace equation

Δu(x)=0 for x ∈ Ω,

u(x)=g(x) for x ∈ ∂Ω. (4.3.52)

4.3 The Initial Boundary Value Problem for the Heat Equation 113

Proof: We write u(x, t)=u

(x, t)+u

(x, t), where u

and u

both solve the

heat equation, and u

has the correct initial values, i.e.,

(x, 0) = f(x)forx ∈ Ω,

while u

has the correct boundary values, i.e.,

(x, t)=g(x)forx ∈ ∂Ω, t > 0,

as well as

(x, t)=0 forx ∈ ∂Ω, t > 0,

(x, 0) = 0 for x ∈ Ω.

By Lemma 4.2.3, we have

lim

t→∞

(x, t)=0.

Thus, the initial values f are irrelevant, and we may assume without loss of

generality that f ≡ 0, i.e., u = u

One easily sees that q(x, y, t) > 0forx, y ∈ Ω, because q(x, y, t)=0

for all x ∈ ∂Ω, and by (iii) of Deﬁnition 4.3.1, q(x, y, t) > 0forx, y ∈ Ω

and suﬃciently small t>0. Since q solves the heat equation, by the strong

maximum principle q then is indeed positive in the interior of Ω for all t>0

(see Corollary 4.3.1).

Therefore, we always have

∂q

∂ν

(x, y, t) ≤ 0. (4.3.53)

Since q(x, y, t) solves the heat equation with vanishing boundary values,

Lemma 4.2.3 also implies

lim

t→∞

q(x, y, t) = 0 uniformly in

Ω ×

Ω (4.3.54)

(utilizing the symmetry q(x, y, t)=q(y, x, t) from Corollary 4.3.1). We then

have for t

|u(x, t

) − u(x, t

)| =





∂Ω

∂q

∂ν

(x, z, t)g(z)do(z)dt



≤ max

∂Ω

|g|



∂Ω



−

∂q

∂ν

(x, z, t)



do(z)dt

= −max |g|



q(x, y, t)dy dt

= −max |g|



(x, y, t)dy dt

= −max |g|



{q(x, y, t

) − q(x, y, t

)}dy

→ 0fort

→∞by (4.3.54).

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

Thus u(x, t) converges for t →∞uniformly towards some limit function u(x)

that then also satisﬁes the boundary condition

u(x)=g(x)forx ∈ ∂Ω.

Theorem 4.3.2 also implies

u(x)=−



∞



∂Ω

∂q

∂ν

(x, z, t)g(z)do(z)dt.

We now consider the derivatives

∂

∂t

u(x, t)=:v(x, t). Then v(x, t) is a solution

of the heat equation itself, namely with boundary values v(x, t)=0for

x ∈ ∂Ω by Lemma 4.3.5. By Lemma 4.2.3, v then converges uniformly to

0on

Ω for t →∞. Therefore, Δu(x, t) converges uniformly to 0 in

Ω for

t →∞,too.Thus,wemusthave

Δu(x)=0.



As a consequence of Theorem 4.3.5, we obtain a new proof for the solv-

ability of the Dirichlet problem for the Laplace equation on bounded domains

of class C

, i.e., a special case of Theorem 3.2.2 (together with Lemma 3.4.1):

Corollary 4.3.3: Let Ω ⊂ R

be a bounded domain of class C

,andlet

g : ∂Ω → R be continuous. Then the Dirichlet problem

Δu(x)=0 for x ∈ Ω, (4.3.55)

u(x)=g(x) for x ∈ ∂Ω, (4.3.56)

admits a solution that is unique by the maximum principle. 

References for this Section are Chavel [3] and the sources given there.

4.4 Discrete Methods

Both for the heuristics and for numerical purposes, it can be useful to dis-

cretize the heat equation. For that, we shall proceed as in Section 3.1 and

also keep the notation of that section. In addition to the spatial variables, we

also need to discretize the time variable t; the corresponding step size will

be denoted by k. It will turn out to be best to choose k diﬀerent from the

spatial grid size h.

The discretization of the heat equation

(x, t)=Δu(x, t) (4.4.1)

is now straightforward:

4.4 Discrete Methods 115



h,k

(x, t + k) −u

h,k

(x, t)



(4.4.2)

= Δ

h,k

(x, t)



i=1



h,k



,...,x

i−1

+ h, x

i+1

,...,x



− 2u

h,k



,...,x



+ u

h,k



,...,x

− h,...,x





Thus, for discretizing the time derivative, we have selected a forward diﬀer-

ence quotient. In order to simplify the notation, we shall mostly write u in

place of u

h,k

. Choosing

=2dk, (4.4.3)

the term u(x, t) drops out, and (4.4.2) becomes

u(x, t + k)=



i=1



,...,x

+ h,...,x



+ u



,...,x

− h,...,x



. (4.4.4)

This means that u(x, t+k) is the arithmetic mean of the values of u at the 2d

spatial neighbors of (x, t). From this observation, one sees that if the process

stabilizes as time grows, one obtains a solution of the discretized Laplace

equation asymptotically as in the continuous case.

It is possible to prove convergence results as in Section 3.1. Here, however,

we shall not carry this out. We wish to remark, however, that the process

can become unstable if h

< 2dk. The reader may try to ﬁnd some examples.

This means that if one wishes h to be small so as to guarantee accuracy of

the approximation with respect to the spatial variables, then k has to be

extremely small to guarantee stability of the scheme. This makes the scheme

impractical for numerical use.

The mean value property of (4.4.4) also suggests the following semidiscrete

approximation of the heat equation: Let Ω ⊂ R

be a bounded domain. For

ε>0, we put Ω

:= {x ∈ Ω :dist(x, ∂Ω) >ε}. Let a continuous function

g : ∂Ω → R be given, with a continuous extension to

Ω \ Ω

, again denoted

by g. Finally, let initial values f : Ω → R be given. We put iteratively

˜u(x, 0) = f (x)forx ∈ Ω,

˜u(x, 0) = 0 for x ∈ R

\ Ω,

u(x, nk)=



B(x,ε)

˜u(y, (n − 1)k) dy for x ∈ Ω, n ∈ N,

and

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

˜u(x, nk)=



u(x, nk)forx ∈ Ω

g(x)forx ∈ R

\ Ω

n ∈ N.

Thus, in the nth step, the value of the function at x ∈ Ω

is obtained as the

mean of the values of the preceding step of the ball B(x, ε). A solution that is

time independent then satisﬁes a mean value property and thus is harmonic

in Ω

according to the remark after Corollary 1.2.5.

Summary

In the present chapter we have investigated the heat equation on a domain

Ω ∈ R

∂

∂t

u(x, t) − Δu(x, t)=0 forx ∈ Ω, t > 0.

We prescribed initial values

u(x, 0) = f(x)forx ∈ Ω,

and in the case that Ω has a boundary ∂Ω, also boundary values

u(y, t)=g(y, t)fory ∈ ∂Ω,t ≥ 0.

In particular, we studied the Euclidean fundamental solution

K(x, y, t)=

(4πt)

−

|x−y|

and we obtained the solution of the initial value problem on R

by convolution

u(x, t)=



K(x, y, t)f(y)dy.

If Ω is a bounded domain of class C

, we established the existence of the

heat kernel q(x, y, t), and we solved the initial boundary value problem by

the formula

u(x, t)=



q(x, y, t)f(y)dy −



∂Ω

∂q

∂ν

(x, z, t − τ )g(z,τ)do(z)dτ.

In particular, u(x, t)isofclassC

∞

for x ∈ Ω, t>0, because of the cor-

responding regularity properties of the kernel q(x, y, t). The solutions satisfy

a maximum principle saying that a maximum or minimum can be assumed

only on Ω ×{0} or on ∂Ω × [0, ∞) unless the solution is constant. Conse-

quently, solutions are unique. If the boundary values g(y) do not depend on t,

then u(x, t) converges for t →∞towards a solution of the Dirichlet problem

for the Laplace equation