Let $\Omega\subseteq\mathbb{R}^N$ be a bounded and smooth domain. If $z:[0,T]\to L^2(\Omega)$ is a function in $H^1([0,T],L^2(\Omega))$ with the property that $z'(t)(x):=z'(t,x)>0$ a.e. on $\Omega$ for any $t\in [0,T]$ is it true that $z$ is an incresing function wrt the order of $L^2(\Omega)$, i.e. $z(t_1,x)>z(t_2,x)$ for almost all $x\in\Omega$ if $t_1>t_2$?

I did not succeed in finding a counterexample. Moreover I find this type of result being used in research articles on reaction-diffusion systems.
For the definition and some basic properties of $H^1([0,T],L^2(\Omega))$ see here <https://www.uni-ulm.de/fileadmin/website_uni_ulm/mawi.inst.020/abschlussarbeiten/MA_Marcel_Kreuter.pdf>

Is there any concept of Dini derivative for such functions?