On the connectedness and asymptotic behaviour of solutions of reaction–diffusion systems

Abstract

In this paper we consider reaction-diffusion systems in which the conditions imposed on the nonlinearity provide global existence of solutions of the Cauchy problem, but not uniqueness. We prove first that for the set of all weak solutions the Kneser property holds, that is, that the set of values attained by the solutions at every moment of time is compact and connected. Further, we prove the existence and connectedness of a global attractor in both the autonomous and nonautonomous cases. The obtained results are applied to several models of physical (or chemical) interest: a model of fractional-order chemical autocatalysis with decay, the Fitz-Hugh-Nagumo equation and the Ginzburg-Landau equation.

Regularity and structure of pullback attractors for reaction–diffusion type systems without uniqueness

“…First, it is known [16,9,5] that under the assumptions (4.4) and (4.5) the problem (4.3) has at least one weak solution u for every given u 0 ∈ L 2 (O). Let us define φ(t, τ, u 0 ) = {u(t + τ, τ, u 0 )} ∈ P (H), ∀t ∈ R + , τ ∈ R, u 0 ∈ H, (4.9)…”

“…From the Kneser property [16] it follows φ(t, τ, u 0 ) ∈ C(H). The strictness property follows from [16,Lemma 3] and the upper semi-continuity from Remark 2.5 and [16, Lemma 2].…”

“…From the Kneser property [16] it follows φ(t, τ, u 0 ) ∈ C(H). The strictness property follows from [16,Lemma 3] and the upper semi-continuity from Remark 2.5 and [16, Lemma 2]. φ indeed satisfies Definition 2.23 since for each u 0 ∈ H and τ ∈ R we can take R τ u0 = {u(t + τ, τ, u 0 )| t 0 }, which satisfies (H1) and (H2) in Proposition 2.25.…”

“…The most direct example of this system is the classical reaction-diffusion equation (taking d = 1) frequently seen in the literature. Nevertheless, the system (1.1) covers more models, such as the Fitz-Hugh-Nagumo equations considered in Section 4.4, and others stated by [16,15].…”

On quasilinear parabolic equations involving weighted p-Laplacian operators

“…However, most of them have been devoted to the existence of global attractors for semilinear non-degenerate parabolic equations and systems (see, e.g. [14,15,20,22] and references therein). In this paper we prove the existence of a global attractor for a class of quasilinear degenerate parabolic equations in an arbitrary domain.…”