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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 $s$?

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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 $G(0)=0$.

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  • $\begingroup$ Thank you very much for your brief verification! $\endgroup$ – Evan William Chandra Apr 5 at 16:46

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