A new (and optimal) result for the boundedness of a solution of a quasilinear chemotaxis–haptotaxis model (with a logistic source)

Liu, Ling; Zheng, Jiashan; Liu, Yu; Yan, Weifang

doi:10.1016/j.jmaa.2020.124231

Cited by 7 publications

(1 citation statement)

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“…Finally, we are aware there exists a vast literature concerning mathematical analysis for dynamical properties of solutions to a general framework of (1.3) with more complex mechanisms like nonlinear diffusion, porous medium slow diffusion, remodeling effects and generalized logistic source etc, cf. [14,15,20,34,35,28,54,55,56] and the references therein. While, upon comparison, we observe that available results on chemotaxis-/haptotaxis systems (especially, for the minimal case like (1.2) and (1.3)) are fully analogous to their corresponding chemotaxis-only systems obtained upon setting w ≡ 0; phenomenologically, any presence of (even sub-)logistic source is enough to prevent blow-up in ≤ 2D and suitably strong logistic damping prevents blowup in ≥ 3D and further strong logistic damping ensures stabilization to constant equilibrium.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Negligibility of haptotaxis effect in a chemotaxis-haptotaxis model

Jin¹,

Xiang²

2020

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In this work, we study chemotaxis effect vs. haptotaxis effect on boundedness, blow-up and asymptotical behavior of solutions for the following chemotaxis-haptotaxis modelin a smooth bounded domain Ω ⊂ R 2 with χ, ξ > 0, η ≥ 0, τ ∈ {0, 1}, nonnegative initial data (u 0 , τ v 0 , w 0 ) and no flux boundary data. In this setup, it is well-known that the corresponding Keller-Segel chemotaxis-only model obtained by setting w ≡ 0 possesses a striking feature of critical mass blow-up phenomenon, namely, subcritical mass ( Ω u 0 < 4π χ ) ensures boundedness, whereas, supercritical mass ( Ω u 0 > 4π χ ) induces the existence of blow-ups. Herein, for some positive number η 0 , we show that this critical mass blow-up phenomenon stays almost the same in the full chemotaxis-haptotaxis model ( * ) in the case of η < η 0 . Specifically, when Ω u 0 < 4π χ , we first show global existence of classical solutions to ( * ) for any η and, then we show uniform-in-time boundedness of those solutions for η < η 0 ; on the contrary, for any given m > 4π χ but not an integer multiple of 4π χ , we detect 'almost' blowup in ( * ) for any w 0 : more precisely, for any ǫ > 0, we construct a sequence of initial data (u ǫ0 , τ v ǫ0 , w 0 ) with Ω u ǫ0 = m such that their corresponding solutions (u ǫ , v ǫ , w ǫ ) satisfy either (A) or (B); here (A) means, for some ǫ 0 > 0, the corresponding solution (u ǫ 0 , v ǫ 0 , w ǫ 0 ) blows up in finite or infinite time, and (B) means 'almost' (approximate) blow-up in the sense, for all ǫ > 0, that the resulting solutions (u ǫ , v ǫ , w ǫ ) exist globally and are uniformly bounded in time but lim infwith some positive and bounded quantity O(1) which can be made explicit. As a result, in the limiting case of ξ = 0, the alternative (A) must happen, coinciding with the well-known supercritical mass blow-up in the chemotaxis-only setting. Also, as a byproduct, in the limiting case of χ = 0, no finite time blow-up can occur for any mass and any η.For negligibility of haptotaxis on asymptotical behavior, we show that any global-in-time w solution component vanishes exponentially as t → ∞ and any global bounded (u, v) solution component converges exponentially to that of chemotaxis-only model in a global sense for suitably large χ and in the usual sense for suitably small χ.Therefore, the aforementioned critical mass blow-up phenomenon for the Keller-Segel chemotaxisonly model is almost undestroyed even with arbitrary introduction of w into ( * ), showing almost negligibility of haptotaxis effect compared to chemotaxis effect in terms of boundedness, blow-up and long time behavior in the chemotaxis-haptotaxis model ( * ).

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