A quasilinear parabolic model for population evolution

Abstract

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 orderpreserving 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.Mathematics subject classification (2010): 35B40, 35K59, 35K92, 92D25.

On maximum and comparison principles for parabolic problems with the p-Laplacian

“…x ∈ Ω and for all s ∈ R + . More detailed weak comparison results for problem (EP) can be found in A. Derlet and P. Takáč [12]. Further results on the existence, uniqueness, and long-time asymptotic behavior of weak solutions are established in [12,33].…”

Nonuniqueness and multi-bump solutions in parabolic problems with the p-Laplacian

“…0.1, p. 552], one obtains also the convergence in C 1+γ,(1+γ )/2 ( × [τ, T ]) for any τ ∈ (0, T ). It is proved in [7] that u is a weak solution to problem (1.1) in the sense of [8, §3.1, p. 23] (cf. Definition 1.1) and it satisfies u ≤ u ≤ u.…”

“…Thus, we have constructed a monotone increasing sequence of subsolutions u = u 1 ≤ u 2 ≤ · · · ≤ u n ≤ · · · to problem (1.1) bounded above by the supersolution u. Standard regularity and compactness arguments from [7] now guarantee that the sequence {u n } ∞ n=1 converges uniformly in × [τ, T ], for any τ ∈ (0, T ), to a continuous function u: × (0, T ] → R. Since also u n (x, 0) ≡ 0 for each n = 1, 2, 3, . .…”

“…We apply the well-known monotone iteration method from Sattinger [20] as adapted to parabolic problems with the p-Laplacian in Derlet and Takáč [7]. In fact, for ordinary differential equations, the method of monotone iterations dates back to the work of Osgood [17].…”

On a comparison method to reaction-diffusion systems and its applications to chemotaxis

“…where µ 1 and µ 2 represent the growth rates of species u and v; the terms −µ 1 u 2 and −µ 1 v 2 represent the inhibition effects that u and v have on the growth of u and v, respectively; the term −µ 1 a 1 uv measures the influence of v on the growth of u; and −µ 2 a 2 vu the inhibiting effect of u on the growth of v. In (7) A i (for i = 1, 2) are second order differential operator of Leray-Lions type (see [6]) defined by 2) are strongly monotone in the following interpolation sense: there exist θ i ∈ (0, 1] and c > 0 such that for all φ 1 , φ 2 ∈ V i we have…”