I am studying a problem of the form $$i \partial_t \psi(t) = L \psi(t) + \int_0^t U(t-r) \psi(r) dr, \qquad \psi(0) = \psi_0,$$ where the evolution occurs in some Hilbert space, $L$ is a self-adjoint differential operator and $U(\cdot)$ is a time-dependent family of differential operators. I would like to know if the initial condition given above is sufficient for well-posedness (or at least, to determine a solution) or if one requires more information.

In particular, any references to papers/books where these types of integro-differential PDEs have been studied would be appreciated. I am calling this thing "non-local" as the time derivative on the left hand side is not "local" in the sense that it does not depend only on information about the solution at time $t$, but depends on the solution from $0$ to $t$. Thank you!

I am studying a problem of the form $$i\, \partial_t \psi(t) = L \psi(t) + \int_0^t U(t-r) \psi(r) \, dr, \qquad \psi(0) = \psi_0,$$ where the evolution occurs in some Hilbert space, $L$ is a self-adjoint differential operator and $U(\cdot)$ is a time-dependent family of differential operators. I would like to know if the initial condition given above is sufficient for well-posedness (or at least, to determine a solution) or if one requires more information.

In particular, any references to papers/books where these types of integro-differential PDEs have been studied would be appreciated. I am calling this thing "non-local" as the time derivative on the left hand side is not "local" in the sense that it does not depend only on information about the solution at time $t$, but depends on the solution from $0$ to $t$. Thank you!