Large time behavior of solution to quasilinear chemotaxis system with logistic source

Abstract

This paper deals with the quasilinear parabolic-elliptic chemotaxis system ut = ∇ • (D(u)∇u) − ∇ • (χu∇v) + µu − µu r , x ∈ Ω, t > 0, τ vt = ∆v − v + u, x ∈ Ω, t > 0, under homogeneous Neumann boundary conditions in a bounded domain Ω ⊂ R n with smooth boundary, where τ ∈ {0, 1}, χ > 0, µ > 0 and r ≥ 2. D(u) is supposed to satisfy D(u) ≥ (u + 1) α with α > 0. It is shown that when µ > χ 2 16 and r ≥ 2, then the solution to the system exponentially converges to the constant stationary solution (1, 1).

A quasilinear predator-prey model with indirect prey-taxis

“…Step 2: With the conclusions in Step1 and refer to [47], we prove the (1.10) by contradiction. First of all, we assume that (1.10) is false, then we can find λ > 0, (t k ) k∈N ∈ (1, ∞) and (x k ) k∈N ∈ Ω such that for all k ∈ N, which contradicts to (4.13).…”

Global boundedness and asymptotic behavior of solutions for a quasilinear chemotaxis model of multiple sclerosis with nonlinear signal secretion

“…We should mention that recently, some global existence results and large time behavior of solutions have also been established for the classical Keller-Segel model, both for the quasilinear parabolic-parabolic and parabolic-elliptic chemotaxis-growth system with nonlinear signal production, one can refer to the literatures [10,36,38,37]. However, to the best of our knowledge, until now there are no results for stabilization of quasilinear chemotaxis model of multiple sclerosis with nonlinear signal secretion, especially, for the case τ = 0.…”

“…Boundedness for multiple sclerosis chemotaxis model with τ = 0. In this section, based on the ideas of [36] and [37], we proceed to derive the main step towards our boundedness proof with τ = 0. Let us first concentrate on the case that m < 1 + 2 N − l − r 2 .…”

Blow-up prevention by logistic source in an $ N $-D chemotaxis-convection model of capillary-sprout growth during tumor angiogenesis

“…To the best of our knowledge, it is yet unclear whether for N ≥ 3 or m < 3, the solution for system (1.1) is existent or not. For more works about the chemotaxisonly models and its variant, we refer readers to see [5,6,14,19,20,21] and the recent survey [1]. Motivated by the above works, we consider system (1.1) without any restriction on the space dimension.…”