Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity

By Youshan Tao, Michael Winkler. 2011

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Abstract We consider the quasilinear parabolic–parabolic Keller–Segel system { u t = ∇ ⋅ ( D ( u ) ∇ u ) − ∇ ⋅ ( S ( u ) ∇ v ) , x ∈ Ω , t > 0 , v t = Δ v − v + u , x ∈ Ω , t > 0 , under homogeneous Neumann boundary conditions in a convex smooth bounded domain Ω ⊂ R n with n ⩾ 1 . It is proved that if S ( u ) D ( u ) ⩽ c u α with α 2 n and some constant c > 0 for all u > 1 , then the classical solutions to the above system are uniformly-in-time bounded, provided that D ( u ) satisfies some technical conditions such as algebraic upper and lower growth (resp. decay) estimates as u → ∞ . This boundedness result is optimal according to a recent result by the second author (Winkler, 2010 [27] ), which says that if S ( u ) D ( u ) ⩾ c u α for u > 1 with c > 0 and some α > 2 n , n ⩾ 2 , then for each mass M > 0 there exist blow-up solutions with mass ∫ Ω u 0 = M . In addition, this paper also proves a general boundedness result for quasilinear non-uniformly parabolic equations by modifying the iterative technique of Moser–Alikakos (Alikakos, 1979 [1] ).

Subfields: Cell Biology, Modeling and Simulation, Molecular Biology

View Paper: https://www.semanticscholar.org/paper/f61a3c825528abff254983dbaddcfdc3906809c2

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Citations in the corpus

Boundedness and large time behavior in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion and general sensitivity (2015, positive): “1 in [26] which by means of a Moser-type iteration applied to the first equation in (2.”
Global existence and boundedness in a 3D Keller–Segel–Stokes system with nonlinear diffusion and rotational flux (2017, background): “On the other hand, for general m ∈ R, the existence of global bounded solutions can be ensured by m + α > 2 − 2 d (see [5,6,16,19,22,27] and references therein).”
Global classical solution and boundedness to a chemotaxis‐haptotaxis model with re‐establishment mechanisms (2018, methodology): “This method has also been used by some other authors, see for example [18].”
Effects of signal-dependent motilities in a Keller–Segel-type reaction–diffusion system (2017, background): “For instance, quite a comprehensive understanding could be achieved in the case when D = D(u) > 0 and χ = χ(u) ≥ 0 are suitably smooth, independent of the signal concentration v = v(x, t) and exclusively depending on the population density u = u(x, t): Then, namely, known results indicate that essen”
Global classical solutions in a two-dimensional chemotaxis-Navier-Stokes system with subcritical sensitivity (2018, methodology): “7 Bounds for nε and ∇cε in L(Ω) Now our knowledge on regularity of uε is sufficient to allow for the derivation of L p bounds for both nε and ∇cε by means of an analysis of a functional combining ∫ Ω n p ε with ∫ Ω |∇cε| with arbitrarily large p and suitably chosen q = q(p), thus following an approa”
Locally bounded global solutions to a chemotaxis consumption model with singular sensitivity and nonlinear diffusion (2016, background): “hm. 6.38] (estimate for ∇u), [23, Thm. 6.39] (existence and uniqueness) and [23, Thm. 6.40] (uniform boundedness). iv) The same theorems as in the proof of i) apply. v) This is (part of) Lemma A.1 in [36]. Lemma 2.2. For every positive function D ∈ C0([0,∞)), for every L > 0 and α ∈ (0,1) there i”
Combining effects ensuring boundedness in an attraction–repulsion chemotaxis model with production and consumption (2022, neutral): “(12) Further, uniform boundedness of (u, v, w) is ensured whenever (boundedness criterion) u ∈ L∞((0, Tmax); L(Ω)), with p > 1 arbitrarily large, for m1,m2,m3 ∈ R, and with p > n2 , for m1 = m2 = m3 = 1 (see, for instance, [1,28]).”