Global and cocycle attractors for non-autonomous reaction-diffusion equations. The case of null upper Lyapunov exponent

Abstract

In this paper we obtain a detailed description of the global and cocycle attractors for the skew-product semiflows induced by the mild solutions of a family of scalar linear-dissipative parabolic problems over a minimal and uniquely ergodic flow. We consider the case of null upper Lyapunov exponent for the linear part of the problem. Then, two different types of attractors can appear, depending on whether the linear equations have a bounded or an unbounded associated real cocycle. In the first case (e.g. in periodic equations), the structure of the attractor is simple, whereas in the second case (which occurs in aperiodic equations), the attractor is a pinched set with a complicated structure. We describe situations when the attractor is chaotic in measure in the sense of Li-Yorke. Besides, we obtain a non-autonomous discontinuous pitchfork bifurcation scenario for concave equations, applicable for instance to a linear-dissipative version of the Chafee-Infante equation.

Forwards attraction properties in scalar non-autonomous linear–dissipative parabolic PDEs. The case of null upper Lyapunov exponent

“…In all what follows we assume that the upper Lyapunov exponent of the linear part of the equations is null. The reference Caraballo et al [3] contains a detailed analysis of the structure of the global and cocycle attractors of these equations with Neumann or Robin boundary conditions, and now we extend the same conclusions to the case of Dirichlet boundary conditions.…”

“…whereas Caraballo et al [3] have studied the structure of the attractor when λ P = 0. Still many interesting problems remain open in the difficult case λ P = 0.…”

“…Since g(p, x, 0) = 0 for any p ∈ P and x ∈Ū , the mapf : P × X → X, (p, z) →f (p, z),f (p, z)(x) = h(p, x) z(x) + g(p, x, z(x)), x ∈Ū is well defined. Arguing exactly as in Caraballo et al [3] (where only Neumann or Robin boundary conditions are considered), we build the family (p ∈ P ) of associated ACPs in the space X,…”

“…Proof. We do not give the proof in detail, since it suffices to follow the ideas in the proof of Theorem 3.1 in [3]. Basically, one approximates the initial problem by a sequence of regular problems whose solutions are classical, so that the standard methods of comparison of solutions (for instance, see Fife and Tang [11]) apply to these problems, and the obtained inequalities are preserved when taking limits.…”

“…Section 2 contains some basic facts in non-autonomous dynamical systems which will be required in the rest of the work. In Section 3 we show that the main dynamical ingredients used in Caraballo et al [3] to describe the structure of the global attractor of the linear-dissipative reaction-diffusion equations with Neumann and Robin boundary conditions remain valid in the study of the attractor in the space X α when the boundary conditions are of Dirichlet type. We also show that the upper Lyapunov exponents of the linear semiflows respectively on P × C 0 (Ū ) and P × X α are the same, and that the restriction of both topologies on the attractor A agree.…”