Universität Basel
Prof. Dr. Enno Lenzmann
Nichtlineare Evolutionsgleichungen
22.04.2015
Problem Set No. 7
Deadline: Due on Wednesday, April 29th, 2015.
Problem 7.1. (12 Points). Let Ω Ă Rd be an open and bounded set. Recall
that C0 pΩq “ tu P CpΩ̄q : u|BΩ “ 0u is a Banach space equipped with the norm
}u} “ supxPΩ |upxq|.
(i) Let g : R Ñ R be locally Lipschitz continuous with gp0q “ 0. Show that
F : C0 pΩq Ñ C0 pΩq with F puqpxq “ gpupxqq is locally Lipschitz continuous.
(ii) Show that, in general, we do not have the inclusions C0 pΩq Ă H01 pΩq or
H01 pΩq Ă C0 pΩq, by giving examples for the interval Ω “ p0, 1q Ă R.
Problem 7.2. (12 Points). Let Ω Ă Rd be an open and bounded set. Define
λ :“
}∇u}2L2 pΩq
inf
1
uPH0 pΩq,uı0
}u}2L2 pΩq
.
It can be shown that λ ą 0 holds (see, e. g., my lecture notes for Variationsrechnung). Suppose that u “ upt, xq is a smooth solution of
$
for t ą 0 and x P Ω,
& Bt u “ ∆u
u“0
for t ě 0 and x P BΩ,
%
u“ϕ
for t “ 0 and x P Ω.
Show the inequality
}uptq}L2 pΩq ď e´λt }ϕ}L2 pΩq
for all t ě 0.
Hint: Find a suitable function f ptq involving λ and }uptq}L2 pΩq .
Problem 7.3. (12 Points). Consider the initial-value problem for the linear heat
equation posed on the interval Ω “ p0, 1q with zero boundary condition:
$
for t ą 0 and x P p0, 1q,
& ut “ uxx ,
upt, 0q “ upt, 1q “ 0,
for t ě 0,
%
up0, xq “ φpxq P C0 pp0, 1qq.
From class we recall that if φpxq ě 0 then upt, xq ě 0 for all t ą 0 and x P Ω. (That
is, the semigroup Sptq for the heat equation is positivity preserving.)
Show that the following fourth-order equation does not have such a positivity
preserving property in general: More precisely, let Ω “ p0, 1q and suppose that
u P C 8 pr0, 8q ˆ Ωq is a smooth solution of
$
for t ą 0 and x P p0, 1q,
& ut “ ´uxxxx ,
upt, 0q “ upt, 1q “ 0,
for t ě 0,
%
up0, xq “ φpxq P C08 pp0, 1qq.
Now take a function φpxq ě 0 such that φpx˚ q “ 0 for some x˚ P p0, 1q and φpxq ą 0
for all x ‰ x˚ sufficiently close to x˚ . (Thus φpxq has a strict minimum at x˚ .)
Show that it possible that Bt upt, x˚ q ă 0 for t close to t “ 0 and conclude that
upt, x˚ q ă 0 for t close to t “ 0.
1