ABSTRACT
The main result of this chapter establishes that the dynamic of (2.1) is governed by its maximal non-negative steady state, i.e., by the maximal nonnegative solution of the semilinear elliptic problem{ −d∆u = λu+ a(x)f(x, u)u in D,
u = M on ∂D. (2.2)
In particular, it will be shown that, for every M > 0, (2.2) possesses a unique positive solution, which is a global attractor for the solutions of (2.1). Throughout this book, to be consistent with the notation introduced in Theorem 1.7, this solution will be denoted by θ[λ,D,M ]. One of the main results of this chapter establishes that
lim M↓0
θ[λ,D,M ] =
{ 0 if λ ≤ λ1[−d∆, D], θ[λ,D] if λ > λ1[−d∆, D],
where θ[λ,D] := θ[λ,D,0]
Dynamics
stands for the unique positive solution of{ −d∆u = λu+ a(x)f(x, u)u in D, u = 0 on ∂D.
