Bifurcation results for a class of periodic quasi‐linear parabolic equations
Piero De Mottoni, Andrea Schiaffino, Karl Peter Hadeler · Mathematical Methods in the Applied Sciences · 1981
Abstract We consider the problem where a and f are 1‐periodic in t, a is positive, f satisfies appropriate decreasing conditions; smoothness of a, f, ∂Ω is also assumed. Denote by λ0 the principal eigenvalue of Δ with zero Dirichlet boundary conditions, and define . We prove: (a) if ε ≤ 0, then no non‐negative periodic solution exists but zero, and any solution with continuous non‐negative initial datum converges to zero uniformly as t → ∞; (b) if ε > 0, then a unique non trivial non‐negative 1‐periodic solution u* exists, and any solution with continuous, non‐negative not identically zero initial datum approaches uniformly u* as t → ∞.