My graduation paper was about stability theorems for $C_0$-semigroups (see the Wikipedia article for the definitions: http://en.wikipedia.org/wiki/C0-semigroup). I am about to go in another direction for my master degree paper, namely shape optimisation. But before I leave behind $C_0$-semigroups, I would like to know if there is some aplicability of the stability theorems I know in this field. The only applications I found for my paper were about the Hille-Yosida theorem and some of its applications to existence and uniqueness of solutions of partial differential equations.

I will not put any names to my theorems, since maybe they are not known to the world as my teachers name them. Here are some of them:

The $C_0$-semigroup $\{T(t)\}_{t \geq 0}$ is exponentially stable if and only if there exists $p \geq 1$ such that $\int_0^\infty \|T(t)\|^pdt <\infty$.

The $C_0$-semigroup $\{T(t)\}_{t \geq 0}$ is exponentially stable if and only if it satisfies the following condition: For any $f \in \mathcal{C}$ it follows that $x_f \in \mathcal{C}$ where $x_f: \Bbb{R}_+ \to X,\ x_f(t)=\int_0^t T(t-s)f(s)ds$, and $\mathcal{C} = \{ f : \Bbb{R}_+ \to X,\ f \text{ continuous and bounded } \}$.

The last theorem can be formulated and proved in some cases for $(L^p,L^q)$ spaces with $(p,q) \neq (1,\infty)$. A more general concept, dichotomy can be formulated (the space splits into two spaces, on one of them there is stability, and on the other one there is instability.

All these sound very nice, and have quite beautiful proofs, but are they applicable to some branches of applied math, such as ordinary or partial differential equations, or they are just pure math, and thats it?