Global existence for infinite dimensional ODE

Question

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). $

Answer 1

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 $$u(t)=G(t,s)u_0+\int_s^tG(t,\tau)b(\tau)d\tau \ .$$