Please use this identifier to cite or link to this item: `http://hdl.handle.net/10077/16212`
 Title: Global stability, or instability, of positive equilibria of p-Laplacian boundary value problems with p-convex nonlinearities Authors: Rynne, Bryan P. Keywords: Global stability; positive equilibria; p-Laplacian Issue Date: 2017 Publisher: EUT Edizioni Università di Trieste Source: Bryan P. Rynne, "Global stability, or instability, of positive equilibria of p-Laplacian boundary value problems with p-convex nonlinearities", in: "Rendiconti dell’Istituto di Matematica dell’Università di Trieste. An International Journal of Mathematics", 49 (2017), Trieste, EUT Edizioni Università di Trieste, 2017, pp. 193-206 Journal: Rendiconti dell’Istituto di matematica dell’Università di Trieste: an International Journal of Mathematics Part of: 49 (2017) Abstract: We consider the parabolic, initial value problem vt = Δp(v) + λg(x, v)φp(v), in Ω x (0,∞), v = 0, in ∂Ω x (0,∞), (IVP) v = v0 > 0, in Ω x {0}, where Ω is a bounded domain in RN , for some integer N > 1, with smooth boundary ∂Ω, φp(s) := |s|p−1 sgn s , s ∈ R , and Δp denotes the p -Laplacian, with p > max{2,N} , v0 ∈ C0(Ω) , and λ > 0 . The function g : Ω x [0,∞) → (0,∞) is C0 and, for each x ∈ Ω , the function g(x, ·) : [0,∞) → (0,∞) is Lipschitz continuous and strictly increasing.Clearly, (IVP) has the trivial solution v ≡ 0 , for all λ > 0 . In addition, there exists 0 < λmin(g) < λmax(g) such that:• if λ ∈/ (λmin(g),λmax(g)) then (IVP) has no non-trivial, positiveequilibrium;• there exists a closed, connected set of positive equilibria bifurcatingfrom (λmax(g), 0) and ‘meeting infinity’ at λ = λmin(g) .We prove the following results on the positive solutions of (IVP):• if 0 < λ < λmin(g) then the trivial solution is globally asymptoticallystable;• if λmin(g) < λ < λmax(g) then the trivial solution is locally asymptotically stable and all non-trivial, positive equilibria are unstable;• if λmax(g) < λ then any non-trivial solution blows up in finitetime. Type: Article URI: http://hdl.handle.net/10077/16212 ISSN: 0049-4704 eISSN: 2464-8728 DOI: 10.13137/2464-8728/16212 Rights: Attribution-NonCommercial-NoDerivatives 4.0 Internazionale Appears in Collections: Rendiconti dell’Istituto di Matematica dell’Università di Trieste: an International Journal of Mathematics vol.49 (2017)

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