Bifurcation and hyperbolicity for a nonlocal quasilinear parabolic problem

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.