Math295b Winter Term 2002

Assignment 2

1. Prove that the following inequality

u(t) ≤ ( max_{0≤τ≤t} α(τ)) e^{∫_{0}^{t} β(τ) dτ} , t ≥ 0

is valid for any choice of non negative continuous functions α, β and u satisfying

u(t) ≤ α(t) + ∫_{0}^{t} β(τ)u(τ) dτ , t ≥ 0 .

A C_{0}-semigroup T on a Banach space E is called analytic if it allows for an analytic strongly continuous extension to a sector Σ_{δ} = [arg(z) < δ] of the complex plane for some δ ∈ (0, π/2], that is, if

(i) T (0) = id_{E} , T (z_{1} + z_{2}) = T (z_{1})T (z_{2}) , z_{1}, z_{2} ∈ Σ_{δ} .

(ii) T : Σ_{δ} → L(E) is analytic.

(iii) lim_{Σ_{δ}∋z→0} T (z)x = x for all x ∈ E .

It can be shown that the above conditions are equivalent to

(i) T (t)E ⊂ dom(A) , t > 0 .
(ii) ∥tAT (t)∥_{L(E)} ≤ c < ∞ , t > 0 .

where A : dom(A) ⊂ E ⟶ E is the generator of T.

2. Let A : dom(A) ⊂ E ⟶ E be the generator of an analytic C_{0}- semigroup T on E. Let f ∈ C^{ρ}([0, T], E) for some ρ ∈ (0, 1) and show that the mild solution u : [0, T] → E of

u̇ − Au = f(t) , u(0) = x ∈ E ,

given by

u(t) = T (t)x + ∫_{0}^{t} T (t − τ)f(τ) dτ , t ∈ [0, T]

is actually differentiable for t > 0.

3. Let A : dom(A) ⊂ E ⟶ E be defined through

E = L_{2}(0, 1) ,
dom(A) = {u ∈ H^{2}(0, 1) | u(0) = u(1) = 0} ,
Au = ∂_{xx}u , u ∈ dom(A) ,

and show that it generates an analytic C_{0}-semigroup on E.

4. Define T (t) ∈ L(L_{2}(ℝ^{n})) through

\begin{cases}(1 − △)^{−t} = F^{−1}(1 + |ξ|^{2})^{−t}F , & t > 0 \\ id_{L_{2}(ℝ^{n})} , & t = 0 . \end{cases}

Show that T is a C_{0}-semigroup on L_{2}(ℝ^{n}). What is its generator?

5. For a bounded domain Ω ⊂ ℝ^{n} with smooth boundary, for b ∈ L_{∞}(Ω) and c ∈ L_{∞}(Ω) let A be the operator induced by the Dirichlet form

a(u, v) = ∫_{Ω}[(∇u |∇v) + (b |∇u)v + cuv] dx , u, v ∈ H^{∘}^{1}(Ω)

on H^{−1}(Ω). Show that it generates a C_{0}-semigroup on H^{−1}(Ω).

The Homework is due February 15