I have a few questions about a basic differetial equation. Below, there are some of my considerations. I would be very greateful for your help.
Let $f\colon[0,T]\times\mathbb{R}\to \mathbb{R}$ be continuous and Lipschitz with respect to the second argument. Consider the following Cauchy problem: $$u'(t)=f(t,x(t)),\qquad u(0)=u_0.$$ Then, Pickard's theorem gives unique solution with the following regularity $C([0,T])$. I don't want to look at this problem this way anymore. Hence, I introduce the framework of evolution triples $V\subset H\subset V^*$. In what follows, I assume that $f\colon [0,T]\times V\to V$ and for every $v\in V$ $f(\cdot\,,v)\in L^2(0,T,V)$. Consider the following problem:
Find $u\in W^{1,2}(0,T;V,H)$ such that $$u'(t)=f(t,x(t))\quad\text{a.e.}\quad t\in(0,T),\qquad u(0)=u_0.\qquad (*)$$
Suppose that $u$ is a solution to $(*)$. Since $f$ is integrable, we have that $$\int_{0}^{t}u'(s)\,ds=\int_{0}^{t}f(s, u(s))\,ds,$$ hence $$u(t)=u(0)+\int_{0}^{t}f(s, u(s))\,ds.\qquad (**)$$ Now, I want to apply the following proposition:
Proposition. If $f,g\in L^1([0,T],X)$, then the following conditions are equivalent:
a) $f(t)=v+\int_{0}^{t}g(s)\,ds$, for almost all $t\in [0,T]$,
b) $\int_{0}^{t}f(s)\varphi'(s)\,ds=-\int_{0}^{t}g(s)\varphi(s)\,ds$,
c) for every $v^*\in V^*$, $$\frac{d}{dt}\langle v^*, f(\cdot)\rangle_V=\langle v^*, g(\cdot)\rangle_V$$
in the distributional sense on $(0,T)$.
This gives that $u$ is absolutely continuous ($u\in AC^{1,2}([0,T],V)$). Since the spaces $W^{1,2}(0,T;V,H)$ and $AC^{1,2}([0,T],V)$ can be indentified, then, every solution to (*) is of regularity $W^{1,2}(0,T;V,H)$. Also, a solution of $(*)$ is also a solution to $(**)$. The problem is and it's not clear to me, that if $u\in W(0,T;V,H)$, then $u'\in L^2(0,T;V^*)$. So, now $u'(t)\in V^*$ while from $(*)$ $u'(t)\in V$. I think, using the framework of evolution triples, we may understand it two ways - it depends on what we want ??? What is the regularity of $t\to \int_{0}^{t}f(s,u(s))\,ds$?
Conversly, suppose that $u$ is a solution to $(**)$. Then, $u$ is absolutely continuous and hence differentiable almost everywhere. So every solution of $(**)$ solves also $(*)$. Once again, the regularity of the solution is $W^{1,2}(0,T;V,H)$.
Having this, do you think I can repeat fixed point argumentation from Pickard's theorem and prove the existance of $(*)$ adding some hypothesis on $f$?