Let
- $H$ be a separable $\mathbb R$-Hilbert space
- $(\mathcal D(A),A)$ be a linear operator on $H$
- $(e_n)_{n\in\mathbb N}\subseteq\mathcal D(A)$ be an orthonormal basis of $H$ with $$Ae_n=\lambda_ne_n\;\;\;\text{for all }n\in\mathbb N\tag 1$$ for some $(\lambda_n)_{n\in\mathbb N}\subseteq(0,\infty)$ with $$\lambda_{n+1}\ge\lambda_n\tag 2\;\;\;\text{for all }n\in\mathbb N$$
- $$\mathcal D(A^\alpha):=\left\{x\in H:\sum_{n\in\mathbb N}\lambda_n^{2\alpha}\left|\langle x,e_n\rangle_H\right|^2<\infty\right\}$$ and $$A^\alpha x:=\sum_{n\in\mathbb N}\lambda_n^\alpha\langle x,e_n\rangle_He_n\;\;\;\text{for }x\in\mathcal D(A^\alpha)$$ for $\alpha\in\mathbb R$
- $$e^{-tA}x:=\sum_{n\in\mathbb N}e^{-\lambda_n t}\langle x,e_n\rangle_H e_n\;\;\;\text{for }t\ge 0\text{ and }x\in H$$ and $$S(t):=e^{-tA}\;\;\;\text{for }t\ge 0$$
Moreover, let
- $T>0$
- $u_0\in H$
- $f:H\to H$ with $$\left\|f(x)\right\|_H\le L(1+\left\|x\right\|_H)\;\;\;\text{for all }x\in H\tag 3$$ and $$\left\|f(x)-f(y)\right\|_H\le L\left\|x-y\right\|_H\;\;\;\text{for all }x,y\in H\tag 4$$
It's well known that there is a unique $u\in C^0([0,T],H)$ with $$u(t)=S(t)u_0+\int_0^tS(t-s)f(u(s))\:{\rm d}s\;\;\;\text{for all }t\in[0,T]\tag 5\;.$$ I want to show that $$u(t)\in\mathcal D(A^{1/2})\;\;\;\text{for all }t\in(0,T]\tag 6\;.$$
Let $\alpha\ge 0$. Then, $$S(t)x\in\mathcal D(A^\alpha)\;\;\;\text{for all }t>0\text{ and }x\in H\tag 7$$ with $$\left\|A^\alpha S(t)\right\|_{\mathfrak L(H)}\le\frac C{t^\alpha}\;\;\;\text{for all }t>0\tag 8$$ for some $C\ge 0$. We should be able to prove $(6)$ using $(7)$ and $(8)$. However, the proof that I've found is quite sloppy. They say that $$A^{1/2}\int_0^tS(t-s)f(u(s))\:{\rm d}s=\int_0^tA^{1/2}S(t-s)f(u(s))\:{\rm d}s\;\;\;\text{for all }t\in(0,T]\tag{10}\;.$$ While $A^\alpha$ is closed for all $\alpha\in\mathbb R$, I don't think that $(10)$ holds until we know that $$\int_0^tS(t-s)f(u(s))\:{\rm d}s\in\mathcal D(A^{1/2})\;\;\;\text{for all }t\in(0,T]\tag{11}$$ (Please compare with an answer by Pedro to an other question) and $(11)$ is more or less what we're trying to prove.
So, how can $(5)$ be proven rigorously?