Increasing Powers in a Degenerate Parabolic Logistic Equation

Abstract

Abstract. The purpose of this paper is to study the asymptotic behavior of the positive solutions of the problemwhere Ω is a bounded domain and b(x) is a nonnegative function. We deduce that the limiting configuration solves a parabolic obstacle problem, and afterwards we fully describe its long time behavior.

Uniqueness for an Obstacle Problem Arising from Logistic-Type Equations with Fractional Laplacian

“…Problem of this type arises in the study of asymptotic behaviour, as p → ∞, of logistic type equations (see [10,27] for the case α = 2 and [20] for equations with α ∈ (0, 2)). Problem (1.1) with α = 2 also arises as the limit of some predator-prey models (see [8,11]).…”

Asymptotics for logistic-type equations with Dirichlet fractional Laplace operator

“…The most interesting part is the convergence as p → +∞ because by the known results for the usual Laplace operator (see [4,5,8,11,37]) it is reasonable to expect that the limit function is a solution of some free boundary problem (or, equivalently, the obstacle problem). This phenomenon was studied for the first time by Boccardo and Murat [4] in the case of equations with Leray-Lions type operator and with a = 0, b = 1.…”

“…This phenomenon was studied for the first time by Boccardo and Murat [4] in the case of equations with Leray-Lions type operator and with a = 0, b = 1. Asymptotics of solutions of equations of type (1.1) with classical Laplacian and general a, b was investigated in [11,37]. To our knowledge, there are no asymptotics results for (1.1) with α ∈ (0, 2) when p → ∞.…”

“…In the paper, we combine the methods used in the case of local operators with some new methods based on the probabilistic potential theory and stochastic analysis, which in particular allow us to weaken the assumptions adopted in [11,37]. In the paper, we focus on the nonlocal case (α ∈ (0, 2)), but our proofs apply after obvious changes to the local case (α = 2).…”

A Diffusive Logistic Equation With Memory in Bessel Potential Spaces

“…This model has been widely utilised for many different purposes. See, for example, [5,6,8,9,11,13,14,16,17] and the references therein.…”