Math295b Winter Term 2002

Assignment 1

1. Let T be a c_{0}-semigroup on a Banach space E. Prove that there exist constants M ≥ 1 and ω ∈ ℝ such that

∥T (t)∥_{L(E)} ≤ M e^{ωt} .

2. For a c_{0}-semigroup T on a Banach space E define

ξ(t, x) = T (t)x , t ∈ [0, ∞) , x ∈ E

and prove that the following are equivalent:

(i) ξ(·, x) is differentiable. (ii) ξ(·, x) is right differentiable.

3. Show that the translation semigroup T on BUC(ℝ) defined through

T (t)f(·) = f(· − t) , f ∈ BUC(ℝ)

is strongly continuous and compute its generator.

4. Let A ∈ G(E), x ∈ E and f ∈ C^{1−}([0, ∞) × E, E) and prove that

\begin{cases}u̇ + Au = f(t, u) , t > 0 \\ u(0) = x \end{cases}

has a unique local mild solution u(·, x) ∈ C([0, t^{+}(x)), E) for some t^{+}(x) > 0. [Hint: Use Banach Fixed-Point Theorem combined with the Variation-of-Constants-Formula]

5. Let A ∈ ℂ^{n×n} and show that

e^{−tA} = \frac{1}{2πi} ∫_{∂𝔹(0,R)} e^{λt}(λ + A)^{−1} dλ ,

where R > 0 is such that σ(−A) ⊂ 𝔹(0, R) and the integration is counterclockwise. [Hint: Show that both sides of the equation satisfy the same ODE and prove that they have the same initial value by reducing the problem to the case where A is a Jordan block]

The Homework is due February 1 2002