# Proof of Proposition 5.2.3 An introduction to semilinear evolution equations / Thierry Cazenave and Alain Haraux

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

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

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 $$$$\forall t \in [0,T], u(t) = \mathscr{T}(t)\phi + \int_{0}^{t}\mathscr{T}(t-s)F(u(s))ds \tag{5.4}$$$$

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!