The logistic equation with nonlinear advection term. (June 2022) - Cintra, Willian; Montenegro, Marcelo; Suárez, Antonio
Abstract: We study the logistic equation with nonlinear advection term − Δ u + α → ⋅ ∇ u p = λ u − u 2 in Ω with u = 0 on ∂ Ω, where Ω ⊂ R N, N ≥ 1, is a bounded domain with smooth boundary, α → = ( α 1, …, α N ) is a flow satisfying suitable condition, p > 1 and λ ∈ R . The classical logistic equation differs from ours by the inclusion of the nonlinear advection term α → ⋅ ∇ u p, meaning that the velocity of transport movement of the species depends on its density and may be higher or lower, depending on u and the size of p and | α → | . From the mathematical point of view, the inclusion of this term brings technical difficulties in the analysis, especially because it has no definite sign. Depending on λ, by the method of subsolution and supersolution and Picone Identity we show the existence and uniqueness of a positive solution. We also study the behavior of the solutions with respect to α → and as p → 1 + . Highlights: A new equation modeling the behavior of a species is proposed. In this model a nonlinear convective term is included, that means that the species has a directed motion and that the rate at which the population moves up depends on the density of the species. We show the existence and uniqueness of positive solution. Moreover, if the rate of advection movement is very large, the species will not survive.