Boundedness in the higher-dimensional chemotaxis–haptotaxis model with nonlinear diffusion

Abstract

We consider the chemotaxis-haptotaxis model ⎧ ⎨in a bounded smooth domain ⊂ R n (n ≥ 2), where χ , ξ and μ are positive parameters, and the diffusivity D(u) is assumed to generalize the prototype D(u) = δ(u + 1) −α with α ∈ R. Under zero-flux boundary conditions, it is shown that for sufficiently smooth initial data (u 0 , v 0 , w 0 ) and α < 2 n − 1, the corresponding initial-boundary problem possesses a unique global-in-time classical solution which is uniformly bounded. This paper develops some L p -estimate techniques and thereby extends boundedness results in n ≤ 3 to arbitrary space dimensions.

Boundedness of the solution of a higher-dimensional parabolic–ODE–parabolic chemotaxis–haptotaxis model with generalized logistic source

“…Now, applying almost exactly the same arguments as in the proof of Lemma 3.2 of [34] (the minor necessary changes are left as an easy exercise to the reader), we conclude the following Lemma:…”

“…Global existence, boundedness and asymptotic behavior of solution were studied in [15,17,32,46]. [16,34,35,51]). …”

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

“…The following Gagliardo‐Nirenberg inequality (see Wang and Winkler and Djie) will be used in deriving the L p estimates of u ε and |∇ v ε |.…”

“…The conclusion is that large values of m seem to enhance the tendency towards the global solvability. When m > 2 − 2/ n , Wang further obtained the existence of global‐in‐time solutions for the system . It is worthy of mentioning that the diffusion coefficient in their studies was nonlinear but eventually assumed to be nondegenerate.…”

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.…”