I am currently studying part II of the monograph "Superlinear parabolic problems: Blow-up, Global Existence and Steady States" by Quittner/Souplet.
In chapter 17 the model problem $$ \left\lbrace\begin{matrix} u_t - \Delta u = \lambda u - \vert u \vert^{p-1} u & x \in \Omega, t >0, \\ u = 0 & x \in \partial\Omega, t>0 \\ u(x,0) = u_0 (x) & x \in \Omega \\ \end{matrix} \right. $$ is considered, where $p>1, \lambda \in \mathbb{R}$. Then the related the energy functional $$ E(u) = \frac{1}{2} \int_\Omega \vert \nabla u\vert^2 - \lambda u^2 \, \mathrm{d}x - \frac{1}{p+1} \int_\Omega \vert u\vert^{p+1} \, \mathrm{d}x $$ is studied in order to derive a criterion for Blow-up of solutions. In Lemma 17.5 the authors prove that for $u_0 \in H_0^1 (\Omega) \cap L^\infty (\Omega)$ it holds: $E(u(\cdot)) \in C([0,T)) \cap C^1 ((0,T))$ by explicitly computing $$ \frac{\text{d}}{\text{d}t} E(u(t)) = - \int_\Omega (u_t (t))^2 \, \text{d}x. $$ I am a bit stuck in understanding the proof at a crucial point: It is claimed that $u \in C((0,T), H^2 (\Omega)) \cap C^1 ((0,T), L^2 \cap L^{p+1} (\Omega))$ is sufficient to perform the following passage to the limit: For $s\neq t$ in $(0,T)$ $$ -\int_\Omega \frac{u(t) - u(s)}{t-s} \, \Delta (u(t)+ u(s)) \, \text{d}x \overset{s \rightarrow t}{\longrightarrow} -2\int_\Omega u_t (t) \, \Delta (u(t) \, \text{d}x. $$ I think this can be justified by using the Dominated Convergence Theorem, but I don't know how to choose the dominating function. Moreover, I don't find a rigorous definition of the spaces $C((0,T), X)$, $C^1 ((0,T), Y)$. Do you know a citable reference for that?
Sorry for the wall of text! Any help is welcome :)
Best regards!
Edit: Typos...