A critical exponent in a degenerate parabolic equation

We consider the chemotaxis-fluid system          n t + u · ∇n = ∇ · (D(n)∇n) − ∇ · (nS(x, n, c) · ∇c), c t + u · ∇c = ∆c − nf (c),in a bounded convex domain Ω ⊂ R 3 with smooth boundary, where φ ∈ W 1,∞ (Ω) and D, f and S are given functions with values in [0, ∞), [0, ∞) and R 3×3 , respectively.In the existing literature, the derivation of results on global existence and qualitative behavior essentially relies on the use of energy-type functionals which seem to be available only in special situations, necessarily requiring the matrix-valued S to actually reduce to a scalar function of c which, along with f , in addition should satisfy certain quite restrictive structural conditions.The present work presents a novel a priori estimation method which allows for removing any such additional hypothesis: Besides appropriate smoothness assumptions, in this paper it is only required that f is locally bounded in [0,for all n ≥ 0 with some k D > 0 and someIt is shown that then for all reasonably regular initial data, a corresponding initial-boundary value problem for (0.1) possesses a globally defined weak solution.The method introduced here is efficient enough to moreover provide global boundedness of all solutions thereby obtained in that, inter alia, n ∈ L ∞ (Ω × (0, ∞)). Building on this boundedness property, it can finally even be proved that in the large time limit, any such solution approaches the spatially homogeneous equilibrium (n 0 , 0, 0) in an appropriate sense, where n 0 := 1 |Ω| Ω n 0 , provided that merely n 0 ≡ 0 and f > 0 on (0, ∞). To the best of our knowledge, these are the first results on boundedness and asymptotics of large-data solutions in a three-dimensional chemotaxisfluid system of type (0.1).

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“…In view of the Gagliardo-Nirenberg inequality (see [31] for a version involving integrability exponents less than one) and (2.4), we can find…”

Section: Proofmentioning

confidence: 96%

Boundedness and large time behavior in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion and general sensitivity

2015

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“…This role will be played by the following semi-convexity estimate which is a well-known feature of nonlinear diffusion equations of type (1.5). For its derivation, we may thus refer to the literature (see [1] or [23], for example).…”

Section: Upper Bounds Formentioning

confidence: 99%

Slow growth of solutions of superfast diffusion equations with unbounded initial data

Фила

Winkler

2017

J. London Math. Soc.

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“…Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php Table 1 Solution behavior in the parameter ranges from Figure 1 in the case when large time limit. Indeed, some semilinear and quasi-linear parabolic equations are known to allow for phenomena of this type (cf., e.g., [27,9,33,34]), but in most examples this kind of behavior seems to be unstable with respect to either the initial data or parameters in the equation. This also applies to some related results addressing the standard parabolic-elliptic Keller-Segel system, where global unbounded solutions are known to exist if the total mass of cells precisely attains some critical value (cf., e.g., [19] for a detailed analysis of the time asymptotics of such solutions).…”

Section: An Illustration Of Parameter Regimes For α and β Which Inmentioning

confidence: 99%

Global Regularity versus Infinite-Time Singularity Formation in a Chemotaxis Model with Volume-Filling Effect and Degenerate Diffusion

Wang¹,

Winkler²,

Wrzosek³

2012

SIAM J. Math. Anal.

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Abstract.A system of quasi-linear parabolic and elliptic-parabolic equations describing chemotaxis is studied. Due to the assumed presence of a volume-filling effect it is assumed that there is an impassable threshold for the density of cells. This assumption leads to singular or degenerate operators in both the diffusive and the chemotactic components of the flux of cells. We improve results from earlier works and find critical conditions which reflect the interplay between diffusion and chemotaxis and warrant that classical solutions are global in time and separated uniformly from the threshold. In the case of degenerate diffusion for the elliptic-parabolic version of the model we prove the existence of radially symmetric solutions which exhibit a phenomenon of infinite-time singularity formation in that they are global and smooth but attain the threshold in the large time limit.

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A critical exponent in a degenerate parabolic equation

Abstract: SUMMARYWe consider positive solutions of the Cauchy problem in R n for the equationand show that concerning global solvability, the number q = p+1 appears as a critical growth exponent.

Cited by 86 publications

References 9 publications

Boundedness and large time behavior in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion and general sensitivity

Boundedness and large time behavior in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion and general sensitivity

Slow growth of solutions of superfast diffusion equations with unbounded initial data

Global Regularity versus Infinite-Time Singularity Formation in a Chemotaxis Model with Volume-Filling Effect and Degenerate Diffusion

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