88 2. FOUR IMPORTANT LINEAR PDE

for a solution of the initial/boundary-value problem

⎧

⎨

⎩

ut − uxx = 0 in R+ × (0, ∞)

u = 0 on R+ × {t = 0}

u = g on {x = 0} × [0, ∞).

(Hint: Let v(x, t) := u(x, t) − g(t) and extend v to {x 0} by odd

reflection.)

16. Give a direct proof that if U is bounded and u ∈ C1

2(UT

) ∩ C(

¯T

U )

solves the heat equation, then

max

¯

U

T

u = max

ΓT

u.

(Hint: Deﬁne uε := u − εt for ε 0, and show uε cannot attain its

maximum over

¯

U

T

at a point in UT .)

17. We say v ∈ C1

2(UT

) is a subsolution of the heat equation if

vt − Δv ≤ 0 in UT .

(a) Prove for a subsolution v that

v(x, t) ≤

1

4rn

E(x,t;r)

v(y, s)

|x −

y|2

(t − s)2

dyds

for all E(x, t; r) ⊂ UT .

(b) Prove that therefore max

¯

U

T

v = maxΓT v.

(c) Let φ : R → R be smooth and convex. Assume u solves the heat

equation and v := φ(u). Prove v is a subsolution.

(d) Prove v :=

|Du|2

+ ut

2

is a subsolution, whenever u solves the

heat equation.

18. (Stokes’ rule) Assume u solves the initial-value problem

utt − Δu = 0 in

Rn

× (0, ∞)

u = 0, ut = h on

Rn

× {t = 0}.

Show that v := ut solves

vtt − Δv = 0 in

Rn

× (0, ∞)

v = h, vt = 0 on Rn × {t = 0}.

This is Stokes’ rule.

19. (a) Show the general solution of the PDE uxy = 0 is

u(x, y) = F (x) + G(y)