  In this article, we study a one-dimensional nonlocal quasilinear problem of
the form $u_t=a(\Vert u_x\Vert^2)u_{xx}+\nu f(u)$, with Dirichlet boundary
conditions on the interval $[0,\pi]$, where $0<m\leq a(s)\leq M$ for all $s\in
\mathbb{R}^+$ and $f$ satisfies suitable conditions. We give a complete
characterization of the bifurcations and of the hyperbolicity of the
corresponding equilibria. With respect to the bifurcations we extend the
existing result when the function $a(\cdot)$ is non-decreasing to the case of
general smooth nonlocal diffusion functions showing that bifurcations may be
pitchfork or saddle-node, subcritical or supercritical. We also give a complete
characterization of hyperbolicity specifying necessary and sufficient
conditions for its presence or absence. We also explore some examples to
exhibit the variety of possibilities that may occur, depending of the function
$a$, as the parameter $\nu$ varies.

Analysis of PDEs;Dynamical Systems