No abstract

Inertial manifold theory, saddle point property and exponential dichotomy have been treated as different topics in the literature with different proofs. As a common feature, they all have the purpose of 'splitting' the space to understand the dynamics. We present a unified proof for the inertial manifold theorem, which as a local consequence yields the saddle-point property with a fine structure of invariant manifolds and the roughness of exponential dichotomy. In particular, we use these tools in order to establish the hyperbolicity of certain global solutions for non-autonomous parabolic partial differential equations.

In this paper we study the asymptotic behaviour of solutions for a non-local non-autonomous scalar quasilinear parabolic problem in one space dimension. Our aim is to give a fairly complete description of the forward asymptotic behaviour of solutions for models with Kirchhoff type diffusion. In the autonomous case we use the gradient structure, symmetry properties and comparison results to obtain a sequence of bifurcations of equilibria, analogous to what is seen in the local diffusivity case. We provide conditions so that the autonomous problem admits at most one positive equilibrium and analyse the existence of sign changing equilibria. Also using symmetry and the comparison results (developed here) we construct what is called non-autonomous equilibria to describe part of the asymptotics of the associated non-autonomous non-local parabolic problem.

In this article, we study a one-dimensional nonlocal quasilinear problem of the form u t = a( u x2 )u xx + νf (u), with Dirichlet boundary conditions on the interval [0, π], where 0 < m ≤ a(s) ≤ M for all s ∈ 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(•) 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 ν varies.