Boundedness in a chemotaxis–haptotaxis model with nonlinear diffusion

Abstract

This article deals with an initial-boundary value problem for the coupled chemotaxishaptotaxis system with nonlinear diffusionunder homogeneous Neumann boundary conditions in a bounded smooth domain Ω ⊂ R n , n = 2, 3, 4, where χ, ξ and µ are given nonnegative parameters. The diffusivity D(u) is assumed to satisfy D(u) ≥ δu m−1 for all u > 0 with some δ > 0. It is proved that for sufficiently regular initial data global bounded solutions exist whenever m > 2 − 2 n . For the case of non-degenerate diffusion (i.e. D(0) > 0) the solutions are classical; for the case of possibly degenerate diffusion (D(0) ≥ 0), the existence of bounded weak solutions is shown.Math Subject Classification (2010): 35K55 (primary), 35B40, 92C17

A new result for 2D boundedness of solutions to a chemotaxis–haptotaxis model with/without sub-logistic source

“…For convenience, we collect the widely used well-known Gagliardo-Nirenberg inequality. [14,21,22]) Let Ω ⊂ R n (n ≥ 1) be a bounded smooth domain and let p ≥ 1 and q ∈ (0, p). Then there exists a positive constant C GN = C(p, q, n, Ω) such that…”

“…for some given nonnegative parameters ǫ, η, τ, α, β, γ, δ, χ, ξ, µ and m ∈ R. For effects of possible interactions between various mechanisms on dynamical properties of the corresponding IBVP for model (1.8) and its variants, we refer the interested reader to [15,19,22,25,29,30,31,32,33,37,40,43,48,61,62,63].…”

Global classical solution and boundedness to a chemotaxis-haptotaxis model with re-establishment mechanisms

“…The stability of the constant stationary solution also be established by Tao and Winkler for and large (compared with the chemotactic coefficient ). Besides these, there are also some papers concerned with the nonlinear diffusion, see for example . When , becomes the haptotaxis‐only system, the global solvability and boundedness were studied in for the 2D and 3D cases.…”

Global existence of solutions to a chemotaxis‐haptotaxis model with density‐dependent jump probability and quorum‐sensing mechanisms

“…To account for this biological feature, various taxis models with degenerate diffusion have been paid more attention during the last decades. They describe the model for chemosensitive movement, moving towards the gradient of nondiffusible signals (haptotaxis), or incorporating both chemotaxis and haptotaxis effect . Particularly, in Li and Lankeit, it is proved that, for sufficiently regular initial data, the bounded solutions of time globally exist for the cases of nondegenerate diffusion and degenerate diffusion whenever m > 2−2/ n with n = 2, 3, 4.…”

“…Since boundedness of u in L ∞ (Ω) is evident in the case when M j ≤maxf∫ Ω ðu 0 þ 1Þ r j þ1 dx; 1g for infinitely many j⩾1, we may assume that M j ≥maxf∫ Ω ðu 0 þ 1Þ r j þ1 dx; 1g and thus, according to (14), there holds…”