Let $(X, ||\,\cdot\,||)$ be a Banach Space and its norm. We want to look for $T>0$ and $u$ a solution of the following problem : $\begin{cases} u \in C([0,T],D(A))\cap C^{1}([0,T],X) \\ u'(t) = Au(t) + F(u(t)), \forall t \in [0,T] \\ u(0) = x \end{cases} $. Here, $A$ is $m$-dissipative operator with dense domain and $(T(t))_{t\geq 0}$ is the contraction semigroup generated by $A$. Morever, $F$ is Lipschitz with Lipschitz constant $L(\,\cdot\,)$ which depends on the size of a ball centered at $0$ with radius $(\,\cdot\,)$ denoted by $B_{(\,\cdot\,)}$.
$D(A)$ here denotes Banach Space $(D(A), ||\,\cdot\,||_{D(A)})$ with $||u||_{D(A)} = ||u|| + ||Au||$.
Now, we want to reformulate the problem above into the following problem: \begin{align}\tag{1} u(t) = T(t)x + \int_{0}^{t}T(t-s)F(u(s))ds, \, \, \forall t \in [0,t] \end{align} Before I mention the part in which I do not understand, I would like to mention some important related lemmas, theorems, and propositions.

Note : $L(\,\cdot\,)$ is a non-decreasing function with respect to $(\,\cdot\,)$.

Lemma 4.3.2
Let $T > 0, x\in X$, and $u,v \in C([0,T],X)$ be two solutions to problem (1). Then $u = v$.

Set $T_{M} := \frac{1}{2L(2M+||F(0)||)+2}>0$ and $T(x) := \sup\{T>0 \,|\, \exists u \in C([0,T],X) \text{ solution of (1)}\}$. Then, we have the following proposition.
Proposition 4.3.3
Let $M>0$ and let $x\in X$ be such that $||x||\leq M$. Then, there exists a unique solution $u \in C([0,T_{M}],X)$ of (1) with $T = T_{M}$.

Theorem 4.3.4 There exists a function $T : X \to (0,\infty]$ with the following properties :
$\forall x \in X, \exists u \in C([0,T(x)),X) \ni 0<\forall T<T(x), u \text{ is the unique solution of }(1)\text{ in }C([0,T],X)$
Moreover, \begin{align}\tag{2} \forall t \in [0,T(x)) \quad 2L(||F(0)||+2||u(t)||) \geq \frac{1}{T(x)-t}-2 \end{align} In particular, we have the following alternatives:
(i) $T(x) = \infty$;
(ii) $T(x) < \infty$ and $\lim\limits_{t\uparrow T(x)} ||u(t)|| = \infty$.

Now, this is the proposition in which I do not understand.
Proposition 4.3.7
Following the notation of Theorem 4.3.4, we have the following properties :
(i) $T : X \to (0,\infty]$ is lower semicontinuous.
(ii) $x_{n}\to x$ and $T< T(x)$ $\implies u_{n} \to u$ in $C([0,T],X)$ where $u_{n}$ and $u$ are the solutions of (1) corresponding to the inidital data $x_{n}$ and $x$ respectively.

So, the proof says that it is enough to show that if $(x_{n})_{n\geq 0} \subset X$ is a sequence s.t. $x_{n}\to x$ as $n\to \infty$, then for $n$ sufficiently large $T(x_{n}) >T$ and $u_{n} \to u$ in $C([0,T],X)$. I do not understand why this also shows that $T$ is lower semicontinuous.

Also, let us define $M := 2 \sup\limits_{t\in [0,T]} ||u(t)||$ and $\tau_{n} := \sup \{ t \in [0,T(x_{n})) \,|\, \forall s \in [0,T] \, ||u_{n}(s)|| \leq 2M \}$. The book says for $n$ large enough, $||x_{n}||< M$ and so $\tau_{n} > T_{M} > 0$. I understand $||x_{n}||< M$ but I do not see how and why $\tau_{n} > T_{M}$ here. Any clue or guidance is pretty much appreciated here!

For reference, this is from section 4.3.2 (Continuous dependence on initial data) of "An Introduction to Semilinear Evolution Equations" by T. Cazenave, A. Haraux, and Y. Martel.


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