  We consider several models (including both multidimensional ordinary
differential equations (ODEs) and partial differential equations (PDEs),
possibly ill-posed), subject to very strong damping and quasi-periodic external
forcing. We study the existence of response solutions (i.e., quasi-periodic
solutions with the same frequency as the forcing). Under some regularity
assumptions on the nonlinearity and forcing, without any arithmetic condition
on the forcing frequency $\omega$, we show that the response solutions indeed
exist. Moreover, the solutions we obtained possess optimal regularity in
$\varepsilon$ (where $\varepsilon$ is the inverse of the coefficients
multiplying the damping) when we consider $\varepsilon$ in a domain that does
not include the origin $\varepsilon=0$ but has the origin on its boundary. We
get that the response solutions depend continuously on $\varepsilon$ when we
consider $\varepsilon $ tends to $0$. However, in general, they may not be
differentiable at $\varepsilon=0$. In this paper, we allow multidimensional
systems and we do not require that the unperturbed equations under
consideration are Hamiltonian. One advantage of the method in the present paper
is that it gives results for analytic, finitely differentiable and low
regularity forcing and nonlinearity, respectively. As a matter of fact, we do
not even need that the forcing is continuous. Notably, we obtain results when
the forcing is in $L^2$ space and the nonlinearity is just Lipschitz as well as
in the case that the forcing is in $H^1$ space and the nonlinearity is $C^{1 +
\text{Lip}}$. In the proof of our results, we reformulate the existence of
response solutions as a fixed point problem in appropriate spaces of smooth
functions.
