Abstract
A quasilinear parabolic problem is investigated. It models the evolution of a single population species with a nonlinear diffusion and a logistic reaction function. We present a new treatment combining standard theory of monotone operators in L 2 (Ω) with some order- preserving properties of the evolutionary equation. The advantage of our approach is that we are able to obtain the existence and long-time asymptotic behavior of a weak solution almost simultaneously. We do not employ any uniqueness results; we rely on the uniqueness of the minimal and maximal solutions instead. At last, we answer the question of (long-time) survival of the population in terms of a critical value of a spectral parameter.
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