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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. 
   $X = C_{0}(Omega) := {fin C(overline{Omega}),|,f|_{partialOmega}=0}$ 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. 
   $(S(t))_{tgeq 0}$ 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, $F(,.,):Xto X$ 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 $nablaphi in L^{2}(Omega)$, 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 $phi in Xcap H_{0}^{1}(Omega)$, 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} + Cint_{0}^{t}frac{1}{sqrt{t-s}}||F(u(s))||ds \
&leq ||S(t)phi - phi||_{H^1} + Csqrt{t} to 0 text{ as }tdownarrow 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 $uin C([0,T],X)$. 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 $(mathscr{T}(t))_{tgeq 0} = (S(t))_{tgeq 0}$ 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!