Slowly non-dissipative equations with oscillating growth

Abstract

The goal of this paper is to construct explicitly the global attractors of semilinear parabolic equations when the reaction term has an oscillating growth. In this case, solution can also grow-up, and hence the attractor is unbounded and induces a flow at infinity. In particular, we construct heteroclinic connections between bounded and/or unbounded hyperbolic equilibria when the reaction term is asymptotically linear.

Autonomous and Non-autonomous Unbounded Attractors in Evolutionary Problems

“…I − P)S(s)u 0 − (P S(s)u 0 ) ≤ M u 0 + 3MC f γ 2 e −δs for s ≥ 0, and in consequence for every B ∈ B(X ) there exists a constant C(B) > 0 such that dist (S(s)B, graph ) ≤ C(B)e −δs .Proof By(21) then there exists δ > 0 such thatξ(t) ≤ M ξ(τ ) e −δ(t−τ ) for t ≥ τ.By the definition of ξ(•) we deduce(I − P)v(t) − * (t, Pv(t)) ≤ M (I − P)v(τ ) − * (τ, Pv(τ )) e −δ(t−τ ) .But, exploring the relation between the function v, the solution of(24), and the solution of the original problem, we obtain(I − P)u(t) − (I − P)u(t) − * (t, Pv(t)) ≤ M (I − P)u(τ ) − (I − P)u(τ ) − * (τ, Pv(τ )) e −δ(t−τ ) . and the bound in the definition of LB(κ) we deduce(I − P)S(t − τ )u(τ ) − (I − P)u(t) − * (t, Pv(t)) ≤ M u(τ ) + 3MC f γ 2 e −δ(t−τ )Take t = 0 and τ = −s for s ≥ 0.…”

“…The set I is the maximal invariant set if I = {x ∈ X : there is a global solution γ : R → X through x which is bounded in the past} It is also possible to define another notion, which coincides with the classical concept in the theory of global attractors. Note that in Example 1 I \ I b consists of heteroclinics to infinity as defined by [6], see also [18] and the articles [3,13,21,27].…”

“…Finally, in Sect. 2.7, we discus, on an abstract level, the dynamics at infinity, which corresponds to the equilibria at infinity and their connections in [3,6,18,21,27], and we prove, using the Hopf lemma, and homotopy invariance of the degree, that the attractor at infinity always coincides with the whole unit sphere in E + .…”

“…The next series of our results is inspired by the works of [6,18] later extended and refined in [3,21,27]. Namely, for the case of the semigroup governed by the following PDE in one space dimension u t = u x x + bu + f (u, u x , x), on the space interval, with appropriate (typically Neumann) boundary data, and sufficiently large constant b > 0 the authors there study the detailed structure of the unbounded attractor which consists of equilibria, their heteroclinic connections, appropriately understood 'equilibria at infinity' and their connections, as well as heteroclinic connections from the equilibria to 'equilibria at infinity'.…”

“…such that the orbits are not absorbed by one given bounded set, and they possibly diverge to infinity when time tends to infinity, but there is no blow-up in finite time. We remark, that in recent years, stemming from [9], there appeared significant work in the framework of unbounded attractors, cf., [3,6,13,18,21,27]. An abstract approach to autonomous and non-autonomous unbounded attractors has been very recently proposed in [4].…”