Hille-Yosida existence theorem theorem is one example of abstract evolution equation. In its simplest form it starts with a closed, densely defined operator $A$ on a Hilbert space $H$ with domain $D(A)$ such that

$$(Au, u)\geq 0,\;\;\forall u\in D(A) $$

and $\boldsymbol{1}+ A: D(A)\to H$ is surjective. (Such operators are called linear *maximal monotone* operators.) The operator $A$ is typically unbounded. We can make $D(A)$ into a Banach space with norm

$$\Vert u\Vert_A:= |u|+ |Au|,\;\;\forall u\in D(A), $$

where $|-|$ denotes the norm in $H$.

The Hille-Yosida theorem states that for any $u_0\in D(A)$ there exists a unique function

$$ u\in C^1\bigl(\;[0,\infty), \; H\;\bigr)\cap C^0\bigl(\; [0,\infty), \;D(A) \;\bigr)$$

satisfying the initial value problem

$$\frac{du}{dt}+ Au=0,\;\; u(0)=u_0. $$

If additionally $A$ is symmetric, i.e., $(Au,v)=(u,Av)$, $\forall u,v\in D(A)$, then one can relax the assumptions on $u_0$. For more details see H. Brezis' book **Functional Analysis, Sobolev spaces and Partial Differential Equations**.

How hard is it to apply this in practice? The surjectivity of $ \boldsymbol{1}+ A$ is the great obstactle because it is in essence an existence result: for any $v\in H$ one can find $u\in D(A)$ such that

$$u+ Au= v. $$

This is nontrivial and requires typically elliptic theory. The domain $D(A)$ of $A$ is often difficult to identify. If for example $A$ is an elliptic operator of the form

$$A= -\sum_{1\leq j} \partial_{x_j}\bigl( a_{ij}(x) \partial_{x_i} \bigr), $$

where $a_{ij}(x)$ are smooth functions on a bounded domain with smooth boundary, then the domain of $A$ depends on the boundary condition you impose. For a Dirichlet condition the domain will be

$$ D(A)= \bigl\lbrace u\in H^2(\Omega);\;\;u|_{\partial \Omega}=0\bigr\rbrace. $$

For more general elliptic operators, the domains could be more complicated. Let me point out that the fact that $A$ defined by the above differential operator with the above domain is a *closed* operator is already a nontrivial fact. It ultimately relies on a priori elliptic estimates for elliptic boundary value problems. In their complete generality they represent a major mathematical achievement of the 20th century.

One can work on Banach spaces as well, and more importantly, the Hille-Yoshida theorem has a *nonlinear* and highly nontrivial generalization. On this topic I recommend V. Barbu's book **Nonlinear Differential Equations of Monotone Types in Banach Spaces** for general abstract nonlinear results and many applications.