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total points: 20
Prof. Dr. Moritz Kaßmann Fakultät für Mathematik
Sommersemester 17 Universität Bielefeld
Partial Differential Equations Exercise sheet VIII, 06/07/17
Exercise VIII.1 (2+3 Points) a) Prove the following lemma:
Lemma. Let T > 0 and v ∈ C([0, T]) nonnegative. Assume there are constants a, b≥0 such that for all t∈[0, T]
v(t)≤a ˆt
0
v(s)ds+b.
Then
v(t)≤beat ≤b(1 +at eat) for all 0≤t≤T.
b) Let Ω ⊂ Rd be a bounded domain with ∂Ω ∈ C1, T > 0 and QT = (0, T]×Ω.
Assume u∈C(QT)∩C1,2(QT)solves
∂tu−∆u=f inQT, u(0,·) =g onΩ,
u= 0 on[0, T]×∂Ω.
for some given continuous functions f andg.
Use part a) to show that for allt∈(0, T]the following estimate holds:
||u(t,·)||2L2(Ω)+ 2||∇u||2L2(Qt)≤et
||g||2L2(Ω)+||f||2L2(Qt)
Exercise VIII.2 (5 Points)
LetB1(0)⊂Rd andQ= (0,1]×B1(0). Show that the following claims are false:
Claim 1: Letu be a positive solution of the heat equation
∂tu−∆u= 0 inQ and t >0.
Then there is a constant C >0, independent of u, such that u(t, x)≤Cu(t, y)
for allx, y∈B1/2(0).
Claim 2: Letu be a positive solution of the heat equation
∂tu−∆u= 0 in(0,1]×Rd, then there is a constantC >0such that for all 23 < s <1
u(s, x)≤Cu(t, y)
for allx, y∈B1/2(0)and 0< t < 13.
Exercise VIII.3 (3+2 Points)
LetΩ⊂Rd be open,α >0 and p∈[1,∞). Defineu:Rd→R, u(x) =
(0, if x= 0,
|x|−α, else.
Find allα, psuch thatu|Ω has a weak derivative inLp(Ω)and decide whether the function is inW1,p(Ω)for
a) Ω =B1(0)the open unit ball, b) Ω =Rd\B1(0).
Exercise VIII.4 (1+2+2 Points)
Let I ⊂ R be a (not necessarily bounded) interval, f ∈ L1loc(I) and u ∈ W1,p(I), 1≤p≤ ∞. Prove the following statements:
a) If ˆ
I
f(t)ϕ0(t)dt= 0 for all ϕ∈Cc1(I),
thenf is constant almost everywhere.
b) For x0 ∈I define a function v:I →Rvia v(x) =
ˆx
x0
f(t)dt.
Then v∈C(I) andf is the weak derivative ofv.
c) There is a function eu∈C(I)such that
u=ue a.e. onI and
u(x)e −eu(y) = ˆx
y
u0(t)dt for allx, y∈I.
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