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Section 9.14 Exercises

Assume \(T(t)\) is a \(C^0\)-semigroup. Show that \(t\mapsto T(t)\) is continuous for all \(t>0\text{.}\)

Consider the shift semigroup \(T(t) u = u(x+t)\) on
\begin{equation*} C_0(\RR) = \{f\in C(\RR), \lim_{x\to \infty} |f(x) | =0\}. \end{equation*}
Show that \(T(t)\) is a \(C^0\)-semigroup.

Suppose \(T(t)\) and \(S(t)\) have the same generator \(A\text{.}\) Show that the semigroups are identical.

Let \(T(t)\) be a \(C^0\)-semigroup with generator \(A\) on the Banach space \(X\text{.}\) Consider \(\phi\in C_c^\infty(0,\infty) \text{.}\)
  1. Show that for each \(n>0\text{:}\) \(\int_0^\infty \phi(s) T(s) x \;ds \in D(A^n)\) for all \(x\in X\text{.}\)
  2. Use this property to show that \(\bigcap_n D(A^n) \) is dense in \(X\text{.}\)

Consider the two operators \(A\) and \(B\) given as
\begin{equation*} A = \frac{\partial}{\partial x}, \qquad D(A) = \{ u\in H^1(0,1); u(1) =0 \}. \end{equation*}
\begin{equation*} B = \frac{\partial}{\partial x}, \qquad D(B) = \{ u\in H^1(0,1); u(0) =0 \}. \end{equation*}
  1. Show that \(A\) generates a \(C^0\)-semigroup.
  2. Show that \(B\) does not generate a \(C^0\)-semigroup.
  3. Can you give an intuitive explanation why this is the case?

For an analytic semigroup \(T(t)\) with generator \(A\text{,}\) we can define the negative fractional powers of \(A\) for \(\alpha>0\) as
\begin{equation*} (-A)^{-\alpha} := -\frac{1}{2\pi i} \int_\Gamma \lambda^{-\alpha} (\lambda I + A) ^{-1} d\lambda, \end{equation*}
where \(\Gamma\) is a curve connecting \(e^{-i\theta} \infty\) with \(e^{i\theta}\infty\text{,}\) with \(0\) on the one side and \(\sigma (-A)\) on the other. Use a Cauchy-integral argument to show that within this definition we have
\begin{equation*} (-A)^{-1} = -A^{-1}. \end{equation*}

Given a Hilbert space \(H\) with orthonormal basis \(\{\psi_i\}_{i=1,2,\dots} \text{.}\) For a given \(n\in\NN\) the projection operator is defined as
\begin{equation*} P_n u = \sum_{i=1}^n (u, \psi_i) \psi_i . \end{equation*}
  1. Show that \(P_n:H\to H\) is generator of a strongly continuous semigroup of quasi contractions.
  2. Find an explicit representation of the semigroup \(e^{P_n t}\text{.}\)

Use a fixed-point argument to solve the reaction diffusion equation on a smooth domain \(\Omega\subset \RR^n\text{:}\)
\begin{align} u_t \amp=\amp \Delta u + f(u) \qquad \mbox{on}\quad \Omega,\nonumber u(0,x) \amp=\amp u_0(x), u(t,x) \amp=\amp 0 \qquad \qquad \quad \mbox{on} \quad \partial\Omega.\nonumber\tag{9.14.1} \end{align}
We assume
  1. {(A1)} The solution space is
    \begin{equation*} X = C^0([0,T], H_0^1(\Omega)), \end{equation*}
    with some \(T>0\) small enough. The norm on \(H_0^1\) will be denoted by the double line notation
    \begin{equation*} \|u\| = \|u\|_{H_0^1(\Omega)}. \end{equation*}
  2. {(A2)} We know that \(\Delta\) generates a \(C^0\)- semigroup \(T(t)\) with norm
    \begin{equation*} \|T(t)\|\leq 1, \qquad\mbox{for all} \qquad t\geq 0. \end{equation*}
  3. {(A3)} We assume that \(f\) is linearly bounded and Lipschitz continuous, i.e., there exist constants \(c_1, c_2>0\) such that for each \(u,v\in H_0^1\text{:}\)
    \begin{align*} \|f(u) \|\amp\leq\amp c_1(1+\|u\|), \|f(u)-f(v)\| \amp\leq \amp c_2 \|u-v\|. \end{align*}
  1. For a given \(\phi\in X\) write down the definition of the norm of \(\phi\) in \(X\text{.}\)
  2. Define a mild solution of the above problem (Display Mathematics [STRUCT].[NUM]).
  3. For each \(0<t<T\) define an operator \(Q\) on \(H_0^1\) as
    \begin{equation*} v\mapsto Qv := T(t) u_0 + \int_0^t T(t-s) f(v(s,x)) ds. \end{equation*}
  4. Denote \(m:=2 \|u_0\|\) and show that, for \(t\) small enough, there is a radius \(R>0\) such that
    \begin{equation*} Q:B_R(0)\to B_R(0), \end{equation*}
    where \(B_R(0)\) denotes the closed ball of radius \(R\) in \(H_0^1(\Omega)\text{.}\)
  5. Show that, for \(t\) small enough, the map \(Q\) is a \(k\)-contraction in \(B_R(0)\text{.}\) Find this \(k\text{.}\)
  6. Use a Fixed Point Theorem to show the existence of a unique mild solution of (Display Mathematics [STRUCT].[NUM]).