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I am currently reading the book stated in the title and there is a part I do not understand in the proof.

Before I state the proposition, I would like to clarify the general assumptions here.

  1. $\Omega \subset \mathbb{R}^{N}$ is a bounded set with Lipschitz continuous boundary.
  2. $X = C_{0}(\Omega) := \{f\in C(\overline{\Omega})\,|\,f|_{\partial\Omega}=0\}$ and $Y = L^{2}(\Omega)$
  3. $D(B) = \{u \in H_{0}^{1}(\Omega) \,|\, \Delta u \in L^{2}(\Omega)\}$ and $\forall u \in D(B), \, Bu = \Delta u$.
  4. $T(\,.\,) : X \to (0,\infty]$ is a function of maximal existence time of the solution. In this case, this function (has been proven) is lower semi-continuous.
  5. $(S(t))_{t\geq 0}$ is a contraction semigroup generated by $B$.

Finally, I will state some used equations here
\begin{equation} \begin{cases} u \in C([0,T],X)\cap C((0,T],H_{0}^{1}(\Omega))\cap C^{1}((0,T],L^{2}(\Omega)); \\ \Delta u \in C((0,T],L^{2}(\Omega)); &(5.1) \\ u_{t} - \Delta u = F(u), \forall t \in (0,T] &(5.2) \\ u(0) = \phi& (5.3) \end{cases} \end{equation} Here, $F(\,.\,):X\to X$ is Lipschitz continuous function

\begin{equation} ||\nabla u||_{L^2}\leq \frac{1}{\sqrt{2t}}||\nabla \phi||_{L^2}\tag{3.32} \end{equation}

This is the statement of the proposition.

Proposition 5.2.3. Assume that $\phi \in X \cap H_{0}^{1}(\Omega)$. Then, the solution corresponding to (5.1)-(5.3) is in $C[0,T(\phi)),H_{0}^{1}(\Omega))$. Then $u$ corresponding to (5.1)-(5.3) is in $C([0,T(\phi)),H_{0}^{1}(\Omega))$. Suppose further that $\nabla\phi \in L^{2}(\Omega)$, then $u \in C([0,T(\phi)),D(B))\cap C^{1}([0,T(\phi)),L^{2}(\Omega))$.

Now, this is the first statement of the proof :
"Assume that $\phi \in X\cap H_{0}^{1}(\Omega)$, and let $t \in (0,T(\phi))$. Applying (5.2), Proposition 3.16, and (3.31), we obtain \begin{align*}\tag{WHY} ||u(t) - \phi||_{H^1} &\leq ||S(t)\phi - \phi||_{H^1} + C\int_{0}^{t}\frac{1}{\sqrt{t-s}}||F(u(s))||ds \\ &\leq ||S(t)\phi - \phi||_{H^1} + C\sqrt{t} \to 0 \text{ as }t\downarrow 0 \end{align*}"

Before I clarify the part I do not understand, I would also like to state this proposition which might be useful.

Proposition 5.1.1. Let $\phi \in X, T>0$, and $u\in C([0,T],X)$. Then, $u$ is solution of (5.1)-(5.3) if and only if $u$ satisfies \begin{equation} \forall t \in [0,T], u(t) = \mathscr{T}(t)\phi + \int_{0}^{t}\mathscr{T}(t-s)F(u(s))ds \tag{5.4} \end{equation}

Also, I would like to note that $(\mathscr{T}(t))_{t\geq 0} = (S(t))_{t\geq 0}$ here.

Now, this is the part I do not understand.
1. How to obtain (WHY)? I do not understand how to transform (5.2) into that inequality.
2. This one is about "Proposition 3.16". The book has no "Proposition 3.16" and so I am confused which proposition the author refers to here in this case.

Any help is very much appreciated! Thank you very much!

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