The Asymptotic Behavior of the Solutions of Degenerate Parabolic Equations

Abstract

ABSTRACT. Existence of stationarystates is established by means of the method of upper and lower solutions.The structure of the solution set is discussed and a uniqueness property for certain classes is proved by a generalized maximum principle. It is then shown that all solutions of the parabolic equation converge to a stationary state.

Nonnegative solutions for the degenerate logistic indefinite sublinear equation

“…By a similar reasoning to the used in the Proposition 4.3, it can be proved that we can apply Lemma I.1 in [8] to show that the pair (u, u) := (εΨ, Ke) is a sub-supersolution of (1.4), provided that ε and K are sufficiently small and large, respectively. Now, the strong maximum principle shows (4.8), see Lemma 2.1 in [7]. This completes the proof.…”

“…By (4.12), it follows that u n ≥ 0, u n = 0 in A k 0 + . From the strong maximum principle, see again Lemma 2.1 in [7], it follows that…”

“…The existence will be shown in Theorem 4.1. For the uniqueness we would like to point out that we use a change of variable already used in a different context in [30], see also [7] and [12]. …”

“…When λ = 0, L = −∆ and a changes sign, (1.4) was studied in detail in [7]. In this work, the authors proved the existence of nonnegative solutions of (1.4).…”

“…In this work, the authors proved the existence of nonnegative solutions of (1.4). Moreover, they showed that when a − ∞ is small, (1.4) possesses a unique nontrivial solution, see Theorem 2.4 in [7]. However, when a − ∞ is large they showed multiplicity results and the existence of dead cores for the solutions, i.e., regions in Ω where the solutions vanish identically.…”