  The purpose of this paper is to study the mixed Dirichlet-Neumann boundary
value problem for the semilinear Darcy-Forchheimer-Brinkman system in
$L_p$-based Besov spaces on a bounded Lipschitz domain in ${\mathbb R}^3$, with
$p$ in a neighborhood of $2$. This system is obtained by adding the semilinear
term $|{\bf u}|{\bf u}$ to the linear Brinkman equation. {First, we provide
some results about} equivalence between the Gagliardo and non-tangential
traces, as well as between the weak canonical conormal derivatives and the
non-tangential conormal derivatives. Various mapping and invertibility
properties of some integral operators of potential theory for the linear
Brinkman system, and well posedness results for the Dirichlet and Neumann
problems in $L_p$-based Besov spaces on bounded Lipschitz domains in ${\mathbb
R}^n$ ($n\geq 3$) are also presented. Then, employing integral potential
operators, we show the well-posedness in $L_2$-based Sobolev spaces for the
mixed problem of Dirichlet-Neumann type for the linear Brinkman system on a
bounded Lipschitz domain in ${\mathbb R}^n$ $(n\geq 3)$. Further, by using some
stability results of Fredholm and invertibility properties and exploring
invertibility of the associated Neumann-to-Dirichlet operator, we extend the
well-posedness property to some $L_p$-based Sobolev spaces. Next we use the
well-posedness result in the linear case combined with a fixed point theorem in
order to show the existence and uniqueness for a mixed boundary value problem
of Dirichlet and Neumann type for the semilinear Darcy-Forchheimer-Brinkman
system in $L_p$-based Besov spaces, with $p\in (2-\varepsilon ,2+\varepsilon)$
and some parameter $\varepsilon >0$.
