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, basically two different types of attractors can appear, depending on whether the linear coefficient in the equations determines a bounded or an unbounded associated real cocycle. In the first case (the one for periodic equations), the structure of the attractor is simple, whereas in the second case (which happens in aperiodic equations), the attractor is a pinched set with a complicated structure. We describe situations in which 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.