  Let $A,C,P:D(A)\subset X\to X$ be linear operators on a Banach space $X$ such
that $-A$ generates a strongly continuous semigroup on $X$, and $F:X\to X$ be a
globally Lipschitz function. We study the well-posedness of semilinear
equations of the form $\dot{u}(t)=G(u(t))$, where $G:D(A)\to X$ is a nonlinear
map defined by $G=-A+C+F\circ P$. In fact, using the concept of maximal
$L^p$-regularity and a fixed point theorem, we establish the existence and
uniqueness of a strong solution for the above-mentioned semilinear equation. We
illustrate our results by applications to nonlinear heat equations with respect
to Dirichlet and Neumann boundary conditions, and a nonlocal unbounded
nonlinear perturbation.
