Recently I have come across a method known as "variational method" in which we try to establish weak solutions of various boundary value problems involving ordinary derivatives, partial derivatives, and fractional derivatives. The use of Sobolev spaces is the key to finding such solutions in which the Lax Milgram theorem is applied to show the existence and uniqueness of solutions. The general approach can be seen in the 8th chapter of the book *"Functional analysis, Sobolev spaces, and partial differential equations"* (publisher link).

Now I was wondering if a similar approach can be applied to check the weak solutions of an abstract evolution equation whose solutions are Banach space-valued functions, i.e, bounded linear operators. The abstract Cauchy problem is:

$$u'(t)=Au(t), \text{for}\hspace{3pt} t\geq 0$$ $$u(0)=x$$ where $A:D(A)\subset X \rightarrow X$ is the linear operator and the generator of a semigroup and $X$ is a Banach space. So far I am aware of two types of solutions namely classical solution in which $u(0)=x \in D(A)$ and a mild/integral solution in which $u(0)=x\in X$.