Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Existence of bounded uniformly continuous mild solutions on $\Bbb{R}$ of
  evolution equations and their asymptotic behaviour   We prove that $u'= A u + \phi $ has on $\Bbb{R}$ a mild solution $u_{\phi}\in
BUC (\Bbb{R},X)$ (that is bounded and uniformly continuous), where $A$ is the
generator of a $C_0$-semigroup on the Banach space ${X}$ with resolvent
satisfying $||R(it,A)||= O(|t|^{-\theta})$, $|t|\to \infty $, with some $\theta
> 1/2$, $\phi\in L^{\infty} (\Bbb{R},{X})$ and $i\,sp (\phi)\cap \sigma
(A)=\emptyset$. As a consequence it is shown that if ${\Cal F}$ is the space of
almost periodic, almost automorphic, bounded Levitan almost periodic or certain
classes of recurrent functions and $\phi$ as above is such that $M_h
\phi:=(1/h)\int_0^h \phi (\cdot+s)\, ds \in \Cal {F}$ for each $h >0$, then
$u_{\phi}\in \Cal {F}\cap BUC (\Bbb{R},X)$. These results seem new and
strengthen several recent theorems.

19
The categories of this paper are: eess
32
The categories of this paper are: atom-ph
13
The categories of this paper are: quant-ph
26
The categories of this paper are: bayes-an
7
The categories of this paper are: hep-th
3
The categories of this paper are: physics
30
The categories of this paper are: q-alg
8
The categories of this paper are: hep-ex
18
The categories of this paper are: econ
35
The categories of this paper are: mtrl-th
11
The categories of this paper are: hep-lat
27
The categories of this paper are: comp-gas
14
The categories of this paper are: math-ph
17
The categories of this paper are: q-fin
6
The categories of this paper are: astro-ph
20
The categories of this paper are: acc-phys