Global existence for infinite dimensional ODE Let us consider the ODE $\hskip3pt \dot x=F(t,x)\hskip3pt $ in an infinite-dimensional  Banach space $E$, where the flux $F$ is defined and continous from  the whole $\mathbb R\times E$ into $E$. 
(1) Question 1. Assuming
$$
\Vert{F(t,x)}\Vert\le \alpha(t)\Vert{x}\Vert,\quad \text{with $\alpha\in L^1_{loc}(\mathbb R)$},
\tag{$\ast$}
$$
is probably not sufficient for global existence: it should be a variation on J. Dieudonné's counterexamples for infinite dimensional ODEs. 
(2) Question 2. However I do believe that solutions of linear equations (i.e. $F(t,x)=A(t) x$, $A(t)$ bounded endomorphism of $E$, depending continuously on $t$)
do  exist globally. Why? Note that it is of course obvious for a constant $A$, since we have in this case the  explicit solution
$
e^{tA}x(0).
$
 A: The answer to (2) is yes, and it's very standard: the corresponding $F$ satisfies the Cauchy-Lipschitz-Picard-Lindelöf hypotheses: being continuous $A$ is locally bounded, so $F$ is locally uniformly Lipschitz in the variable $x$. In fact, for a continuous $A:[a,b]\to L(E)$ you can directly solve the Cauchy problem for the ODE with parameter: 
$$\begin{cases}G(s,s)=I \\ \partial_1 G(t,s)=A(t)G(t,s)   \end{cases}$$
solving the equivalent integral equation
$$ G(t,s)- \int_s^t A(\tau)G(\tau,s)d\tau=I \ , $$
which consists in inverting a quasinilpotent perturbation of the identity on the Banach space $C^0([a,b]\times [a,b],L(E))$. This is easily done in terms of the Neumann series:
$$ G(t,s)=\sum_{k=0}^\infty W_k(t,s) , $$
$$\begin{cases}W_0(t,s)=I \\ W_{k+1}(t,s)=\int_s^t A(\tau)W_k(\tau ,s)d\tau   \end{cases}\ .$$
The variation of constant formula holds too, and produces  the unique solution of 
the non-homogeneous problem 
$$\begin{cases}u(s)=u_0 \\ \dot u(t)=A(t)u(t)+b(t)   \end{cases}$$
as 
$$u(t)=G(t,s)u_0+\int_s^tG(t,\tau)b(\tau)d\tau \ .$$
