Math 295 Winter Term 2008
Assignment 9
1. Let H(t, x) be the heat kernel (the fundamental solution for the heat
equation introduced in class) and show that Tdefined through
(T(t)u(x) := RRnH(t, x −y)u(y)dy , x ∈Rn
T(0)u:= u
is a C0-semigroup of contractions on L2(Rn) but NOT on L∞(Rn).
A C0-semigroup Ton a Banach space Eis called analytic if it allows for an
analytic strongly continuous extension to a sector Σδ= [arg(z)< δ] of the
complex plane for some δ∈(0, π/2], that is, if
(i) T(0) = idE, T(z1+z2) = T(z1)T(z2), z1, z2∈Σδ.
(ii) T: Σδ→ L(E) is analytic.
(iii) lim
Σδ3z→0T(z)x=xfor all x∈E .
It can be shown that the above conditions are equivalent to
(i) T(t)E⊂dom(A), t > 0.
(ii) ktAT (t)kL(E)≤c < ∞, t > 0.
where −A: dom(A)⊂E−→ Eis the generator of T. Show that the
C0-semigroup of problem 1 is analytic.
2. Let −A: dom(A)⊂E−→ Ebe the generator of an analytic C0-
semigroup Ton E. Let f∈Cρ[0, T ], Efor some ρ∈(0,1) and
show that the mild solution u: [0, T ]→Eof
˙u+Au =f(t), u(0) = x∈E ,
given by
u(t) = T(t)x+Zt
0
T(t−τ)f(τ)dτ , t ∈[0, T ]
is actually differentiable for t > 0.
3. Let A: dom(A)⊂E−→ Ebe defined through
E= L2(0,1) ,
dom(A) = u∈H2(0,1) u(0) = u(1) = 0,
Au =−∂xxu , u ∈dom(A),
and show that −Agenerates an analytic C0-semigroup on E.
