Second-Order Parabolic Differential Equations

SECTION

Second-Order Parabolic Differential Equations

We begin the discussion with the solution of the Cauchy problem for the one-dimensional heat equation, that is, the problem (112) φ(χ),

(113)

— oo < x < oo.

We shall look for a solution u(x, t) defined in the half-plane t > 0, assuming bounded and continuous Cauchy data φ(χ) on the x-axis. Theorem 16: If φ(χ) is a continuous bounded function, then the function u(x, t) = —-= f 2ajntJ-«>

(ξ-χ)2

φ(ξ) exp

4a2t

άξ

has derivatives of arbitrary order, with respect to x and t for / > 0. The func­ tion u(x, t) satisfies Eq (112) and lim u(x, t) = φ(χ), ίφθ

for

— oo < x < oo.

Moreover, if u*(x, t) is a solution of (112), satisfying the initial condition (113) with φ*(χ) replacing φ(χ), such that φ*(χ) is continuous and bounded and \φ*(χ)-φ(χ)\ 358


5. SECOND-ORDER PARABOLIC DIFFERENTIAL EQUATIONS

359

then \u*(x, t) — u(x, t)\ < ε for all x and t > 0. PROOF: TO

obtain a solution of Eq. (112), we use the method of separation of

variables : u(x, t) =

X{x)T(t\

This leads to exp[-a2X2Q

T(t) = and

X(x) = A cos λχ + B sin λχ, where λ, A, and B are arbitrary. We may assume that A = Α(λ), B = Β(λ). Since the function ux(x, t) = exp[— α2λ2ί~]\_Α(λ) cos λχ + Β(λ) sin AJC] is a solution of Eq. (112), so is

Γ uk{x9t)dk,

(114)

^-oo

with a suitable choice of A and 5, so as to allow interchanging the order of integration and differentiation. .00

φ(χ) =

u(x, 0) άλ •'-oo

=

-00

(^4(A) cos λχ + B(A) sin Ax) άλ.

•'-00

Since φ(χ) is continuous and bounded, it has a Fourier Integral representation

-oo

v

— oo

A proof of the above result, together with a more detailed discussion of the Fourier transform, will be postponed to Section 6.1. In particular, see Theorem 19(k), p. 382.

360

PARTIAL DIFFERENTIAL EQUATIONS

A comparison of the two representations for φ(χ)9 suggests the following choice for A and B: 1 r00 Α(λ) = — φ(ξ) cos λξ άξ, Β(λ) = ±- Γ

φ(ξ)*ίηλξάξ.

2π J-oo

Substitution of A and B in (114) yields r°° u(x, t) =

wA(x, t) άλ J-αο 00 1 Γ

Λ00

= — dk\ φ(ξ) e x p [ - a 2/ 2i ] cos λ(ξ - x) άξ 2nJ-oo J-o o = - Γ φ(ξ) άξ i ' e x p E - ^ ^ i ] cos λ(ζ - x) άλ \2i

2aJnt 2a-JutJ-«>

L

4a 2 i

άξ

(115)

(see Exercise 1(b)). The function F(x, i; 0 =

7=exP

4a 2 i

considered as a function of x and /, satisfies Eq. (112) for / > 0. Since the integral defining u(x, t) and the integrals obtained from it by differentiation with respect to x and / converge uniformly for \x\ < Λ/, 0 < t0 < t < T, it follows that u(x, t) satisfies (112) for all x, and / > 0. The transformation ξ-χ

2a^t reduces (115) to 1 r°° w(x, t) = —\ φ(χ + 2ayy/t) exp[-.y 2 ] dy. The estimate 1 |n(x, 0 - φ(χ)\ < — y/π'

r00

\φ(χ + 2αγ^ί)

- φ(χ)\ exp[->> 2 ] dy

(116)

5. SECOND-ORDER PARABOLIC DIFFERENTIAL EQUATIONS

361

enables us to show easily that

li (x, t) = φ(χ). lim t

The last part of the theorem follows, similarly, by the estimate \u* x, 0 - u(x, 01 = <

1 la-sjnt

Γ Ιφ*(ξ) - ψ(ξ)-] exp

J

J.

-oo

exp

(x - 0 2 Ί Aa2t

(* - i)2 4a2t

άξ

άξ

The multidimensional heat equation will be discussed in Section 6.2. It should be remarked that the solution w(x, t) of the heat equation is differentiate with respect to the variables x and / independently of the func­ tion φ(χ). Also, u(x, t) > 0 for arbitrary x and t, a fact which physically means that the heat transfer is instantaneous, contrary to the case of the wave equa­ tion. The function F(x, t; ξ) is called the fundamental solution of equation (112). Physically F(x, /; ξ) represents the distribution of temperature due to an instantaneous point source of heat located at ξ. The uniqueness of the soltion of the problem (112), (113) will follow by general maximum and mini­ mum principle for parabolic operators and is to be discussed later in this section. As in the case of elliptic and hyperbolic equations, certain mixed prob­ lems can be solved by the Fourier method (see Exercise 4). We shall conclude this section by proving several general theorems about parabolic operators. We shall limit ourselves to linear normal parabolic equations, in the sense of Definition 2. In particular, we shall assume that n

Lu

= Σ

a

iAX> OUxiXj + Σ bi(X> f)uxt ~ut + C(X9 t)u9

(1.17)

where x = (x1? x 2 , . . . , x„), the coefficients aij9 bt, and c are continuous in a certain region D and the quadratic form Σΐ,ι=ιαυ Pi Pj ^s positive definite in D. Theorem 17: Let L be a parabolic operator in a region D as defined by (117) with c(x, t) = 0 (c < 0) in D. If u e C2(D), Lu > 0 in D and u has a (positive) maximum at a point P0 of D, then u is constant on the set Γ(Ρ0)9 which con­ sists of all the points in D which can be joined to P0 by a line segment per­ pendicular to the /-axis and lying completely in D.

362

PARTIAL DIFFERENTIAL EQUATIONS

PROOF:

(a) We shall first prove the following lemma : If the maximum M of u in D is attained at a point P0 = (x°, t0) which lies on the boundary of a sphere TV contained in D and in which the function u satisfies u < M, then P0 lies on the diameter which is parallel to the i-axis. We may assume that the center of the sphere N is at the origin and that its radius is equal to p. Furthermore, we may assume that P0 is the only maximal point by replacing, if necessary, the original sphere by a smaller sphere, included in the original sphere and tangent to it, at the point P0. If the lemma is false, that is, if x° Φ (0, 0 , . . . , 0), we define hx(x, t) = exp[-2(|x| 2 + t2)-] - e x p [ - V ] One computes Lhx(x, t) = 4λ2 £

aij(x,t)xiXj

2λ Σ bi(x, t)xt + 2Àt + c βχρ[-λ(|χ| 2 + ί 2 )]-αΓ Α '' 2 . (118) If S(P0) is a sufficiently small sphere about the point P0, such that S(P0) c D and (x, t) e S(P0) implies x φ (0, 0 , . . . , 0), then, since the coefficient of λ2 in the brackets of Eq. (118) is positive definite and hence has a positive mini­ mum in S(P0), it follows that the expression in the brackets is positive for large λ in S(P0). Clearly, since c < 0, this implies that Lhk(x91) > 0 for sufficiently large λ and (x, /) e S(P0). Fixing an appropriate A, consider the function v = u + shx,

ε > 0.

On the set dS n N,u < M and 0 < hx < 1, so that for sufficiently small positive ε, v < M on dS n N. On the set dS — N, hx < 0 and u < M, so that also v < M. It follows that v < M on dS, and v(P0) = u(P0) = M. Hence, the function v(x, t) attains its maximum (v attains a positive maximum if c < 0 by the assumption that M > 0) on the set S, at an interior point P'(x\ t'). At this point, we have vx(P') = vt(P') = 09

1=1,2,...,*.

5. SECOND-ORDER PARABOLIC DIFFERENTIAL EQUATIONS

363

It follows that Lv P = P'

= Σ Oijix'OVxtXj i,j=l

+ c(x',t')v(x',t').

(119)

P = P'

On the other hand, the expression Lv = Lu + sLhx is positive in S, since Lu>0 and Lhx > 0 in S. For a fixed / = /', the relation (119) defines an elliptic operator in the variables xl9 x2, . . . , xn in an «-dimensional neighborhood of the point x''. We now obtain a contradiction applying Theorem 13(b) if c < 0 and Corol­ lary 7(a), if c = 0 to the function v^x) = v(x9/') and the set {*| (JC, /') G S}. (b) Next we generalize the lemma in (a) to the case where the sphere N is replaced by an (n + l)-dimensional ellipsoid, one of whose axes is parallel to the /-axis. This follows by applying (a) to a sphere, tangent to the ellip­ soid at the point P0 and contained in it. (c) We now prove the theorem by contradiction. Suppose there exists a point Q e Γ(Ρ 0 ), such that u(P0) > u{Q). Because of the continuity of u(x, /), we may assume that the point P0 is the nearest point on the line segment QP0 at which w(x, /) attains its maximum M. Furthermore, by shifting Q, if necessary, in the direction of P0, we may assume that p(Q, P0) < p(Q, dD). Through Q, we now draw a parallel to the /-axis and mark on both sides two points R and 7, such that | ^ β | = \QT\, p(R, T) < p(Q, P0), RTcz D and u(x, t) < M on the closed interval RT. We now construct a one-parametric family of ellipsoids with center Q, all having n common major axes of length \RT\, one of which is RT. The family is parametrized by the length of the (n + l)th half axis, which is in the direction of P0 Q and its length ranges between zero and p(Q, P0). It is easy to verify that the above family is contained in the sphere of radius p(Q, P0) about Q, and since (Q, P0) < p(Q, dD), is contained in D. Since u < M on the closed segment RT, the same is true in the interior of the ellipsoids of the family with sufficiently small axis in the P0 ß-direction. By a continuity argument, it follows that there exists an ellipsoid of the above family in the interior of which u < M and on at least one point of its boundary, u = M. To this ellipsoid, we can apply the result in (b) and obtain a contra­ diction, since on the diameter 7?!Tparallel to the /-axis u(R) < M and u(T) < M. It can be shown that the strong maximum principle for elliptic operators, that is, Theorem 13(b) follows from Theorem 17 (see Exercise 8). The following are a few corollaries of Theorem 17:

364

PARTIAL DIFFERENTIAL EQUATIONS

Corollary 10: Theorem 17 holds, if the condition Lu > 0 is replaced by Lu < 0, the words, "positive" and "maximum" replaced by "negative" and "mini­ mum," respectively. Corollary 11: Under the assumptions of Theorem 17, u(x, t) is constant in the component of D n {t \ t = t0}, which contains the point P0. This follows immediately by the fact that any two points of the above set can be joined by a polygonal line Next, we prove a considerably stronger version of Theorem 17. Theorem 18 (Nirenberg): Let L be a parabolic operator as in Theorem 17 with c(x,1) = 0, (C(JC, t) < 0). If u e C2(D), Lu>0 in T^PQ) and u has a (positive) maximum with respect to the set T^PQ) at the point P0, then u(x, t) is constant on Τγ(Ρ0). The set Τγ{Ρ), Ρ e D is defined as: T,(P) = {(x, t) | Ms), t(s) e C[0, 1], (x(s), t(s)) e A (x(0), t(0)) = (x, 0, « 1 ) , t(i)) = Λ and t(s) is a nondecreasing function on the interval 0 < s < 1}. The set T^P) can also be described as the set of all points in D that can be joined to P by a Jordan nondecreasing arc. Clearly, if t(s) = t' and x(s), 0 < s < 1, describes the line segment P0 Q, we obtain a nondecreasing Jordan arc joining P0 Q of Theorem 17. Hence, T^PQ) =5 Π^ο). Also, in Theorem 18, we merely assume that P0 is a maximal point, only with respect to T^PQ) and not with respect to the region D. Furthermore, we require Lu>0 only in T^PQ). PROOF:

(a) Assume first, that c = 0. We prove the following: Let R be the rectangle \Xi — jc/|
tl — δ
δ > 0, / = 1, 2, . . . , n,

and contained in Z>; then if the maximum M of u(x, t) in R is attained at the point P1 = (χ', ^ ) , it is also attained for some point in K, with the /-coordinate less than t1. Assume that the above assertion is false, that is, assume that u(x, t) < M in the rectangle Ru where Rx is defined by \xt - x{\
ti— δ
δ > 0.

5. SECOND-ORDER PARABOLIC DIFFERENTIAL EQUATIONS

365

We may assume that Px = {x\ t^) = 0. Define h(x,t) = r2 -\x\2

-(t

+ r)2,

r>0.

Since Lh = -L \x\2 + 2(î + r), it is clear that Lh > 0 in R for sufficiently large r. On the other hand, if ε > 0 is sufficiently small, the function v = u + sh satisfies v < M on dR— {P^. Denote (dR\ = dRn

{(x, t)\ \x\2 + (r + r)2 < r2} - {i\} c Ru

(êR)2 = dR-

(dR)v

For small ε > 0, max

w(x, i) < M;

max

v(x, t) < M.

(x,t)e{dRh

In addition, v(x, t) < M for (x, t) e (dR)2 - {PJ and v{Px) = M. It follows that, for a suitable choice of r and ε, we have Lh > 0 in R,

v
{P^,

and

viPJ = M.

In particular, this implies that Lv > 0 in R. We show next that v attains its maximum over R only on dR. Indeed, assume to the contrary that max υ(χ, t) = v(x*, t*) > v(Pi) = M, (x,t)eK

where (**, /*) e R. Applying Theorem 17 to the function v in the rectangle R, it follows that the function v is constant in the intersection of R and the hyperplane t = t*. Since this set intersects dR at points different from Pu we obtain a contra­ diction. It follows now, that v attains its maximum in R at the point Ργ. This implies that vt(Pt) > 0 and from the relation

ι>,(Λ) = κ,(Λ) + *Αι(Λ) = w ^ ) - 2τε, it follows that ιι,(Λ) > 0.

366

PARTIAL DIFFERENTIAL EQUATIONS

We consider now the region T(R) = Rn

{(x,t)\t

= 0}.

The relation (117), with c = 0 at the point Pi9 has the form n

i, j = l

Hence Σ a0(0,...,OKJP=Pl>0.

(120)

By the continuity of the function in (120), this inequality holds in a neigh­ borhood of Pi with respect to T(R). By Corollary 7(a) applied to the elliptic operator (120) in this neighborhood, we obtain a contradiction to the fact that u(x9 0) attains its maximum at the interior point x = x'. (b) We still assume that c = 0. If Theorem 18 is false, there exists a point Q, Q e T^PQ) such that u(Q) < u(P0) = M. On the Jordan curve joining Q and P0, there is a point P*(x*, t%) such that u(P*) = M and for all other points on the curve joining Q to P*9 u < M. Let t(s)9 0 < s < 1, be the /-coordinate for the curve QP*. If for a certain s = s0, t(s0) = t*, then t(s) = t* for s0 < s < 1, and the corresponding part of the curve would lie in the hyperplane t = t*. By Theorem 17, it would follow that u(P) = M for points P, lying on the Jordan curve between Q and P*, con­ trary to the definition of P*. We can now construct a small rectangle R with upper base on the hyperplane / = t* centered at the point P*9 such that the Jordan curve QP* intersects the lower base at a point Q* between Q and P * and such that Ra D. We have M = u(P*) = max u(x91) (x,f ) e R

and Lu>0

in

K,

since R c rt(P*) c T^PQ). It remains to show that u < M in R^ and to apply part (a) of the proof. Indeed, if u(P) = M,P e Rl9 then by Theorem 17, u = M on the intersec­ tion of R and the hyperplane perpendicular to the /-axis through P. This intersection contains a point between Q and JP* and by assumption, u < M at such a point. This contradiction proves Theorem 18 in the case c = 0.

5. SECOND-ORDER PARABOLIC DIFFERENTIAL EQUATIONS

367

(c) Assume now, c < 0 in D. We can write Lu = L{u + cu, where the parabolic operator Lx satisfies Theorem 18. Since Lu\P=Po = LlU\P=Po

+ c(P0)u(P0) > 0

and u(P0) = M > 0, it is also clear that LlW|P=Po>0.

(121)

Moreover, there exists a neighborhood N(P0) of P0 in which (121) holds because Lu > 0, u(P) > 0 and c < 0, in some neighborhood N(P0). Furthermore, Γ1Ν(Ρο)(/>0) (that is, T^PQ) with respect to N(P0)) is contain­ ed in T^PQ). By Theorem 18 applied to Ll in N(P0), it follows that w(x, i) is constant in ΓίΝ(Ρο)(Ρ0). If our assertion is not true, that is, if there exists a point Q9 Q e T^PQ) such that u{Q) < u(P0), we can also find a point P* on the Jordan arc joining Q to P0 such that w(P*) = M and u(P) < M for all the points on this Jordan arc which lie between Q and P0. Since Γ\(Ρ*) c Γ ^ ο ) , it follows by Theorem 18 applied to L1 and some neighborhood N(P*), that u(x, t) = M in Γ1Ν(Ρ*}(Ρ*). This is impossible, since on the Jordan arc QP*, there are points in N(P*) c Γ ^ ^ ^ Ρ * ) dif­ ferent from P * for which u < M. We shall conclude our discussion with several corollaries. Corollary 12: Theorem 18 holds, if c = 0 and the conditions Lu>0 and "maximum" are replaced by Lu < 0 and "minimum" or if c < 0 and the conditions Lu>0 and "positive maximum" are replaced by Lu<0 and "negative minimum." Corollary 13: If the region D has the property that the regions DT= D n {(x,t)\t<

T}

are bounded for all T, then the maximum of any function u(x, t) e C2(D) n C(D), such that Lu>0 and c = 0 in Z>, with respect to Γ ^ ο ) , Ρ e D is attained on dD. In particular, dD n ^Γ^Ρ) Φ φ for any P e D.

368

PARTIAL DIFFERENTIAL EQUATIONS

If 3D n dT^P) = φ for a certain P e D, then Γ^Ρ) c Z>. Since Γ\(Ρ) is bounded, (Γ^Ρ) is a subset of a set of the form DT where Tis the i-coordinate of P), there exists a point Q e Γ^Ρ) with minimal /-coordinate. A small sphere about Q will still be contained in D and will contain points Px and P2, which can be joined by a Jordan curve with increasing tcoordinate and such that one of the points, say Pi9 has a smaller i-coordinate than Q and the second point P2 e Γ\(Ρ). It would follow that Px e Γ ^ Ρ ) ; this is a contradiction to the definition of Q. Now since u e C(D), we can define PROOF:

max

w(x, 0 = w(P*).

(x,t)ef7(P)

If P * E Γ^Ρ), then u(x, t) is constant in Γ^Ρ*) since Γ^Ρ*) c Γ^Ρ), and hence max

u(x91) =

max

u(x, i) = u(P*);

and thus, Theorem 18 is applicable in Γ\(Ρ*). Since w is continuous in D, u is also constant in Γ ^ Ρ * ) ; and by the previous considerations, this set intersects the boundary of D. If P * e Γ\(Ρ) n Z> - r t ( P ) , then Γ^Ρ*) c f(P) (see Exercise 9). Hence, rt(P*) c Γ(Ρ), and by Theorem 18 and the argument used in the previous paragraph, u(x9 t) is constant in Γ^Ρ*). If P * e dD, this is proof in itself. Corollary 14: Corollary 13 holds if we replace the condition Lu>0 Lu < 0 and the word "maximum" by "minimum."

by

Corollary 15: If u(x9 t) e C2(D) n C(D)9 D is a region as in Corollary 13, then the equation

Lu=f(x9t)

(c = 0),

with boundary conditions

MX 0 U = #(X 0, has, at most, one solution. PROOF:

By Corollaries 13 and 14.

Corollary 16: If Lu = 0, c < 0, then the maximum of \u(x91)\ with respect to Γ^Ρο), where P 0 e D and D is as in Corollary 13, is attained on the boundary of/).

5. SECOND-ORDER PARABOLIC DIFFERENTIAL EQUATIONS

369

Corollary 17: Let L be a parabolic operator with c(x, t) bounded in D and D, as in Corollary 13. The equation Lu = / ( * , 0, with boundary condition u\ôD = 0(x>t),

UEC2(D)Œ

C(D),

has, at most, one solution. PROOF: It is sufficient to show that i f / = 0, the only solution of the above sys­ tem is u ΞΞ 0. If this is not true, let u(x, t) ψ 0 be a solution of the system. Define L* by the relation

L = L* - Dt + c and define the function v(x, t) by the relation v(x, t) = e~Xtu(x, t). We have Lu = L*u — ut + cu = eXtL*v — ek\vt + λν) + cveÀt = eXt(L*v — vt + cv — λν) = e\L

- λ)ν

= 0. Hence (L -λ)ν

=0

and For large λ, c — λ < 0 in D, and by Corollary 16, the maximum of \v(x, t)\ over T^PQ) is attained on the boundary of D but v\dD = 0, hence v = 0 in I^CPo). Since P0 is arbitrary in D, it follows that v = 0 in D. Corollary 18: The equation Au — ut = 0, with the initial condition Φ> 0If = o = / ( * ) ,

370

PARTIAL DIFFERENTIAL EQUATIONS

where/(x) is continuous and bounded for all x e En, has at most one solution u, if u(x91) e C2(t > 0) n C(t > 0) lim u(x, 0 = 0 uniformly on every interval 0 < t < T. We may assume, as before, that / = 0 , If u ψ 0 let u(P0) = w(x°, t0) > 0, 0 < t0 < T(if u < 0 in D we consider the function —u). For sufficiently large K9 we have

PROOF:

|x°| < K and u(x, t) < u(P0),

(122)

for \x\ > K and 0 < t < T. Let (x*, t*) = u(P*) = max w(x, 0 > 0, \x\
and D = {(x, 0 | |*| < K

and

ί > 0}.

Inequality (122) implies that \x*\ < ATand 0 < t+ < T. The set ΓΧ(Ρ*) is T1(P*)={(x,t)\\x\
and

0 < / < /*}.

The function u(x, t) attains its maximum over Γ(Ρ *) at P *. Hence by Theorem 18, w(x, 0 is constant in Γ(Ρ*) and by continuity also in Γ(/ > *). But Γ(Ρ*) includes points (x, i) with t = 0, where w = 0. It is easy to show that the function w(x, /) defined in Theorem 16, satisfies the condition of Corollary 18 if φ(χ)->0 as |x| -► oo (see Exercise 10). It follows that the Cauchy problem for the heat equation has a unique solution. In the case n = 1, this also follows from Theorem 16.

EXERCIS ES

1. (a)

Verify the formula φ{χ) =

1 Λ00

Υηί-

j

ψ(ξ)οο$ίλ(ξ-χ)-]αξ,

5. SECOND-ORDER

PARABOLIC

DIFFERENTIAL

EQUATIONS

371

if φ(χ) is a real-valued step function, assuming only a finite number of distinct values, which vanishes outside some compact set in El9 and x is a point of continuity of φ(χ). (b) Verify the formula

J

-an

exp[ — b2X2~\ cos Ac άλ — ^— exp c D 4b2

for

b > 0.

Hint: Show that the integral of the function exp[ — b2z2~\ exp[/cz] taken over any straight line parallel to the x-axis is constant. Then choose a particular straight line to obtain an integral of the function exp[ — b2k2~\. (c) Verify the representation 2f°° sin λα — sin λχ , „ φ(χ) = αλ, a > 0, 71 Jo

λ

φ(χ) = 1

if \x\ < a and φ(χ) = 0 if \x\ > a. 2. Show that the solution of the problem ut = a2(uxx + uyy + uzz), «lf=o = >%*)>

is given by u(x, v, z, 0 =

-==- ί ί ί φ ( ς , '/, 0 (2α Λ / π 0 x exp —

(è'-x) 2 + (^-y) 2 + (C-^r άξ αη άζ, Â 4a t

3. Solve the problem ut =

a2uxx+f(x,t\

using Duhamel's principle. Verify that u(x, t) = \ dx \

m t) 2ajn(t

- τ)

exp

4α 2 (ί - τ)

άξ.

372

PARTIAL DIFFERENTIAL EQUATIONS

4. Solve by the Fourier method the following: ut = a2uxx+f(x,t), u\t=0 = φ(χ), " l*=o = Φι(0,

0
0
0
\χ=ι =
0
where φ(0) = φ^Ο) and φ(1) = φ2(0). Hint: Consider first the above problems for (a) f=(Pi = (p2 = 0, where φ e Cu φ(0) = φ(1) = 0, and (b) φ = φί = φ2=0,/β C 1 } /(0, t) = / ( / , t) = 0. Then make the substi­ tution u = ι; + φ^Ο + (φ 2 (ί) - φί(ί))(χΙΙ). 5. (a) A homogeneous sphere of radius i^ about the origin is given an initial temperature distribution (r) is a constant. 6. (a) Find the temperature distribution inside an infinite cylinder of radius R in which the initial temperature distribution is given by w0(l — (r2IR2)), suchthat the surface of the cylinder is kept at zero temperature. (b) Solve the same problem assuming the cylinder contains a liquid at temperature 7\ and is submerged in a liquid at temperature T2. 7. Consider the first mixed problem for the heat equation: ut = a\uxx + uyy + uzz\

u li=o = φ(χ, y, z),

(x, y> z) e Ώ,

u\dCl = il/(x,y,z,tl

0
where Ω is a region in the xyz-space. Show directly that if u(x, y, z, t) is a solution of the above problem defined and continuous in QT, QT = Ω χ {t\0 < t m, then show that the function v=u+

2

((x - x0)2 + (y-

y0)2 + (z -

z0)2)

5. SECOND-ORDER PARABOLIC DIFFERENTIAL EQUATIONS

373

does not take a maximum on Γ, hence find a point where vt — a2Av > 0 and obtain a contradiction. 8. Prove Theorem 13(b) assuming Theorem 17. Hint: Assume first c = 0 and apply Theorem 17 to a suitable cylindrical region in the x/-space and the parabolic operator L— Dt9 where L is the given elliptic operator. 9. (a) Show by examples that the sets Γ(Ρ) and T^P) neither open nor closed.

are in general

(b) Prove that P * e YJp) n D implies T^P *) c f\(P), and P * e Γ^Ρ) implies Γ^Ρ*) czT^P). 10. Prove that the function

satisfies l i m ^ i ^ w(x, i) = 0 uniformly in 0 < t < T if limj^^^ φ(ξ) = 0. Here x = (xl9..., χ„), ξ = (ξί9..., ξη). Hint: Use the decomposition $Εη = ί\ξ\<Κ

+

Ι\ξ\>Κ·

11. Solve Exercise 7 using the general theorems of this section. 12. Prove a uniqueness theorem for the solution of the systems in Exercises 4, 5, and 6. 13. Let Ω be a bounded region in the x-space and let QT = Ω x {t\0 < t
where Lw = 0,

c = 0,

and

weC 2 (ß T ).

Prove that if for a certain interval [/1? / 2 ] c= [0, Γ ] , P(0 e QT, then Î;(/) is a nonincreasing function of / in [tu t2~\.