Let $X$ Hilbert space. Let $\mathcal{A}:D(\mathcal A)\subset X\rightarrow X$ nonlinear and $f:[0,+\infty)\rightarrow X$ $$\begin{cases}U_{t}(t)=\mathcal{A} U(t)+f(t),& \forall t>0, \\ U(0)=U_{0}\end{cases} $$ I search for references of theorems which allow to me to have the existence and uniqueness of a solution $U$ in $[0,+\infty)$.
I have found some theorems but in the case where $U$ is defined on $[0,T]$ not on $[0,+\infty)$ and for the inclusion cauchy problem .I mean in this case:
$\mathcal{A}:D(\mathcal A)\subset X\rightarrow X$ nonlinear and $f:[0,T]\rightarrow X$ $$\begin{cases}U_{t}(t)\in\mathcal{A} U(t)+f(t),& \forall t\in[0,T], \\U(0)=U_{0}\end{cases} $$ Remark:
If $X$ Hilbert space et $A:D(A)\subset X \rightarrow X$ nonlinear operator. Cauchy problem is given by : \begin{eqnarray} f(t)&\in&U_t(t)+AU(t) \;t \in [0,T]\\ U(0)&=&U_0 \end{eqnarray} $A$ is$\omega$-m accretive if:
$\langle (A+\omega I) U,U\rangle \geq 0 $ et $R(I+\lambda A)=X $ for some $\lambda >0$, $\omega \in \mathbb{R}^+$.
Theorem:If $A$ is $\omega$-m accretive, $u_0 \in \overline{D(A)}$ and$ f \in L^1(0,T;X)$ then the above Cauchy problem has a unique une solution $ u\in C([0,T];\overline{D(A)})$ (Book "Nonlinear differential equations of monotone types in Banach spaces")
I'd know if there is a reference where I can found this result in $\mathbb{R_+}$ and with equality not inclusion I mean \begin{eqnarray} U_t(t)+AU(t)&=&f(t) \;t \in [0, \mathbb{R}_+)\\ U(0)&=&U_0 \end{eqnarray} if A est $\omega$-m accretive , $u_0 \in \overline{D(A)}$ and $ f \in L^1(\mathbb{R}_+;X)$ then the above Cauchy problem has a unique solution $u\in C(\mathbb{R}_+;\overline{D(A)})$ ???
Are you looking for local existence (ie $U(t)$ and $f(t)$ still defined on $[0,+\infty)$, and there exists $T$ possibly but not necessarily $= + \infty$ such that a unique solution $U(t)$ exists on $[0,T)$?); or global existence (ie existence and uniqueness on entire $[0,+\infty)$)?
For general local existence results, see "Remarks on blow-up and nonexistence theorems for nonlinear evolution equations" (1977) by J.M Ball (https://academic.oup.com/qjmath/article/28/4/473/1563956).
To extend general local existence results to global existence, you will have to prove a continuation theorem (ie norms of the form: $\|\cdot \|_{[0,T],X} < + \infty$ implies solution can be extended locally for $t \geq T$) and show that the such a norm does not blow up in finite time assuming a smooth solution. An example of such a continuation theorem can be found in "Remarks on the breakdown of smooth solutions for the 3-D Euler equations" (1984) by Beale, Kato, and Majda (https://link.springer.com/article/10.1007/bf01212349) for the 3D Euler Equation.