Regularity assumptions needed for the mild representation of the solution to an abstract ODE

Question

Let

• $E$ be a $\mathbb R$-Banach space
• $S:[0,\infty)\to\mathfrak L(E)$ be a $C^0$-semigroup, i.e.
  1. $S(0)=\operatorname{id}_E$
  2. $S(s+t)=S(s)S(t)$ for all $s,t\ge 0$
  3. $t\mapsto S(t)x$ is continuous from $[0,\infty)$ to $E$ for all $x\in E$
• $A$ be the infinitesimal generator of $S$

Let's remind ourselves of a few well-known facts:

Fact 1$\;\;\;$$\mathcal D(A)$ is a dense subspace of $E$.

Fact 2$\;\;\;$$A$ is a closed linear operator from $\mathcal D(A)$ to $E$.

Fact 3$\;\;\;$Let $x\in\mathcal D(A)$ $\Rightarrow$ $$S(t)x\in\mathcal D(A)\;\;\;\text{for all }t\ge 0\tag 1$$ and $$[0,\infty)\to E\;,\;\;\;t\mapsto S(t)x\tag 2$$ is differentiable with $$\frac{\rm d}{{\rm d}t}S(t)x=AS(x)=S(t)A(x)\;\;\;\text{for all }t\ge 0\;.\tag 3$$

Fact 4$\;\;\;$$\mathcal D(A)$ equipped with $$\left\|x\right\|_{\mathcal D(A)}:=\left\|x\right\|_E+\left\|Ax\right\|_E\;\;\;\text{for }x\in\mathcal D(A)$$ is a $\mathbb R$-Banach space.

Now, let

• $T\ge 0$
• $u_0\in\mathcal D(A)$
• $f\in C^0([0,T],E)$
• $u\in C^0([0,T],\color{red}{\mathcal D(A)})\cap C^1([0,T],E)$ with \begin{equation}\left\{\begin{array}{}u'&=&Au+f\\u(0)&=&u_0\end{array}\right.\tag 4\end{equation}
• $t\in[0,T]$ and $$g(s):=S(t-s)u(s)\;\;\;\text{for }s\in[0,t]$$

Using Fact 3 we see that \begin{equation}\begin{split}g'(s)&=-AS(t-s)u(s)+S(t-s)u'(s)\\&=-AS(t-s)u(s)+S(t-s)Au(s)+S(t-s)f(s)\\&=S(t-s)f(s)\end{split}\tag 5\end{equation} and hence $$u(t)-S(t)u_0=g(t)-g(0)=\int_0^tS(t-s)f(s)\;{\rm d}s\tag 6$$

Until now, I haven't equipped $\mathcal D(A)$ with a topology. However, in $(5)$ and $(6)$ we need that $\mathcal D(A)$ is a $\mathbb R$-Banach space such that $$s\mapsto S(t-s)f(s)\tag 7$$ is continuous$^1$ from $[0,t]$ to $E$. If you look up formula $(6)$ in a textbook, the author will state that we need to equip $\mathcal D(A)$ with $\left\|\;\cdot\;\right\|_{\mathcal D(A)}$. The crucial questions are:

1. Why can't we consider $\mathcal D(A)$ as being equipped with the norm induced by $E$? I guess (please correct me) that the (only?) problem is that $\mathcal D(A)$ might not be closed in $E$ (and hence the resulting space wouldn't be a $\mathbb R$-Banach space). Am I right?
2. If we equip $\mathcal D(A)$ with $\left\|\;\cdot\;\right\|_{\mathcal D(A)}$ and want that $(7)$ is continuous, we obviously need that $$s\mapsto S(t-s)\tag 8$$ is continuous from $[0,t]$ to $(\mathcal D(A),\left\|\;\cdot\;\right\|_{\mathcal D(A)})$. Why can we conclude this continuity property from our assumptions?

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$^1$ Among other things, this ensures that the Riemann integral in $(6)$ is defined.

Answer 1

Maybe I didn't understand your considerations, but:

1. If you haven't equipped $\mathcal{D}(A)$ with any topology, then the previous notation $C^0([0,T];\mathcal{D}(A))$ makes no sense.

2. In $(5)$ and $(6)$ you need nothing about $\mathcal{D}(A)$. To calculation $(5)$ be valid (a.e.) you only need assume that $u:[0,T]\to X$ is absolutely continuous and satisfy $(4)$. To get $(6)$ you only need assume that $[0,T]\ni s\mapsto S(t-s)f(s)\in E$ is integrable. Thus, it is enough assume that $f\in L^1(0,T; E)$.

3. In the standard presentations (see my item 2.), a topology for $\mathcal{D}(A)$ plays no role in the derivation of $(6)$. So, give us a reference of such a book please.

4. If your question 1. concerns to the derivation of $(6)$, you don't need consider $\mathcal{D}(A)$ equipped with anything (see my item 3.).

5. With respect to your question 2, again it is not clear for me why you are worrying about $\mathcal{D}(A)$. As I said, in the derivation of $(6)$ you only need $(7)$ integrable as a function with values in $E$ (see my item 2). And you can indeed conclude the regularity of $(7)$ from your previous assumptions (namely, from the assumption on $f$).