I have a question that is somewhat related to PDEs. Take a sequence of functions $\{t\mapsto R_k(t,\cdot)\}_k$ where for each $k$, $R_k(t,x)\in C^{1}(\mathbb{R}_{\geq 0}\times (0,1))$. Let $V$ be a functional regular enough, $V$ could be $\Vert\cdot \Vert^2_{L_2}$ on $(0,1)$. Now define the following sequence of functions with values in $\mathbb{R}$.
$$ \{t\mapsto V_k(t)\}_{k}\colon=\{V(R_k(t;\cdot))\}_k $$ Assume that for each $k$ one has $$ \frac{d}{dt}V_k(t)\leq \mu V_k(t)\qquad t\in [0, T] $$ for some $T$. What I'd like to do is to use the above inequality to conclude something on the behavior of $V_k$ on $[0, T]$ when $k\rightarrow \infty$. In particular, I saw a proof where one shows that under certain assumptions on the limit of the sequence $\{t\mapsto R_k(t,\cdot)\}_k$, the above differential inequality holds in a distributional sense, but I am really confused about all of that. My idea is to get a sort of Lyapunov exponential estimate result for a weak solution to a PDE while assuming the same bound holds for strong solution. I hope my question is somewhat clear.
Thanks!