# Use of L'Hosptial Rule in generalized setting related to heat equation

Let $$X$$ be a space of continuous functions with compact support in a bounded domain $$\Omega \subset \mathbb{R}^{N}$$ with Lipschitz continuous boundary, $$F : X \to X$$ be a Lipschitz continuous function, and $$\varphi \in X$$. Let $$u : [0,T]\times\Omega \to \mathbb{R}$$ be a solution to semilinear parabolic PDE with $$\varphi$$ as the initial condition satisfying : \begin{align*} \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))\\ \forall t\in [0,T],\quad u_{t}(t) - \Delta u(t) = F(u(t))\\ \forall x \in \Omega, \quad u(0,x) = \varphi(x) \end{cases} \end{align*} Let $$(S(t))_{t\geq0}$$ be the contraction semigroup such that the solution satisfies an integral equation $$\forall t\in[0,T], \, u(t) = S(t)\varphi + \int_{0}^{t}S(t-s)F(u(s))ds$$ Moreover, $$S(0) = I$$ is an identity mapping.

Now, I want to ask whether this use of L'Hospital rule can be justified or not in order to calculate $$\lim\limits_{t\to0^{+}}\frac{\int_{0}^{t}S(t-s)F(u(s))-F(\varphi)ds}{t}=0$$ That is, I see that for any fixed $$x \in\Omega$$, I define $$G(t) := \int_{0}^{t}S(t-s)F(u(s))-F(\varphi)ds$$ and thus $$G : [0,T] \to \mathbb{R}$$. Is it justified to use Fundamental Theorem of Calculus here given $$S(t-s)F(u(s))\in X$$ for any fixed $$t$$ and $$s$$?

It is ok if $$G(0) = 0$$. Then it is a $$o/o$$ type limit.
It is true that $$G(0) = u(0,x)-S(0)\phi(x)$$. So from that point of view $$G(0)=0$$.