Global stability, or instability, of positive equilibria of p-Laplacian boundary value problems with p-convex nonlinearities

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, positive equilibrium; • there exists a closed, connected set of positive equilibria bifurcating from (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 asymptotically stable; • 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 finite time