Stability and hyperbolicity of equilibria for a scalar nonlocal one-dimensional quasilinear parabolic problem

Bifurcation and hyperbolicity for a nonlocal quasilinear parabolic problem

“…The proof of hyperbolicity presented here is rather simple compared with that of [2]. Nonetheless, an important part of the analysis is dependent on the analysis done in [2] to view the quasilinear nonlocal problem (1) as a semilinear nonlocal problem. We briefly recall this analysis to take advantage of it.…”

“…For the stationary solutions of (1) we analyze the bifurcations of the equilibria and their hyperbolicity. This analysis has been carried out in [6,2] for the particular case when a is increasing and f is odd (see also [1] for the study of the bifurcation when f is not necessarily odd). These two conditions considerably simplifies the structure of the bifurcations.…”

“…Proceeding as in [2,6], (3) is locally well-posed and the solutions are jointly continuous with respect to time and initial conditions. Changing the time variable to t = τ 0 a( w x (•, θ) 2 ) −1 dθ we have that u(x, t) = w(x, τ ) is the unique solution of (1).…”

“…Let us recall the definitions of local stable and unstable manifolds and the notion of hyperbolicity (see [2]) which applies to (1).…”

“…We refer to this property as topological hyperbolicity. Definition 1.2 (Strict Hyperbolicity, [2]). An equilibrium φ of (1) is said to be hyperbolic if there are closed linear subspaces X u and X s of H 1 0 (0, π) with H 1 0 (0, π) = X u ⊕ X s such that • {φ} is topologically hyperbolic.…”