On the dynamics of a degenerate parabolic equation: global bifurcation of stationary states and convergence

Abstract

We study the dynamics of a degenerate parabolic equation with a variable, generally non-smooth diffusion coefficient, which may vanish at some points or be unbounded. We show the existence of a global branch of nonnegative stationary states, covering both the cases of a bounded and an unbounded domain. The global bifurcation of stationary states, implies-in conjuction with the definition of a gradient dynamical system in the natural phase space-that at least in the case of a bounded domain, any solution with nonnegative initial data tends to the trivial or the nonnegative equilibrium. Applications of the global bifurcation result to general degenerate semilinear as well as to quasilinear elliptic equations, are also discussed.

On quasilinear parabolic equations involving weighted p-Laplacian operators

“…The existence and long-time behavior of solutions to problem (1.1) in the case p = 2, the semilinear case, have been studied in [16,17] and recently in [1]. The aim of this paper is to study the existence and long-time behavior of solutions to problem (1.1) in the case 2 p < N. Noting that the conditions imposed on f provide the global existence of a weak solution to problem (1.1), but not uniqueness.…”

The semiflow of a reaction diffusion equation with a singular potential

“…In Section 3, and in the spirit of our recent work [18], we shall use Theorem 1.1 to discuss the stability properties of equilibria and the asymptotic behavior of solutions of (1.2). We discuss first stability by linearization: Using the improved Hardy inequality of [24] and its consequences, we consider appropriate Garding forms to prove the asymptotic stability of the trivial equilibrium when λ ≤ λ 1,µ and the asymptotic stability of the unique nonnegative equilibrium when λ > λ 1,µ , for 0 < µ ≤ µ * .…”

“…The positivity of u 1,µ follows from [13, Lemma 2.2]-we also refer to the weak maximum principle of [4]. The simplicity and the uniqueness up to positive eigenfunctions of λ 1,µ can be verified, by using Picone's identity [18].…”

“…Proof: The argument of [11,Proposition 5.3.1] for the linear heat equation, can be repeated here (see also [18]). We assume that φ 0 ∈ H µ (Ω), φ 0 ≥ 0 a.e in Ω, and φ(t) = S(t)φ 0 , the global in time solution of (1.2), initiating from φ 0 .…”

Pullback Attractors for a Non-Autonomous Semi-Linear Degenerate Parabolic Equation

“…(H ∞ α,β ) σ satisfies condition (H α ) and lim inf |x|→∞ |x| −β σ (x) > 0 for some β > 2, when the domain is unbounded. For the physical motivation of the assumptions (H α ) and (H ∞ α,β ), we refer the reader to [2,8,14,15].…”