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)
