Asymptotic profile with the optimal convergence rate for a parabolic equation of chemotaxis in super-critical cases

Luckhaus, Stephan; Sugiyama, Yoshie

doi:10.1512/iumj.2007.56.2977

Cited by 39 publications

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“…On the other hand solutions globally exist with small mass and relatively regular initial data, and here the diffusion dominates at large length scale (see [C] and [S2]). Indeed using the entropy dissipation method ( [CJMTU]) it is shown that the solutions with small L 1 and L (2−m)d/2 -norms converge to the self-similar Barenblatt profile ([LS1]- [LS2] and [B2]). …”

Section: Introductionmentioning

confidence: 99%

The Patlak–Keller–Segel Model and Its Variations: Properties of Solutions via Maximum Principle

Kim¹,

Yao²

2012

SIAM J. Math. Anal.

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In this paper we investigate qualitative and asymptotic behavior of solutions for a class of diffusion-aggregation equations. Most results except the ones in section 3 and 6 concern radial solutions. The challenge in the analysis consists of the nonlocal aggregation term as well as the degeneracy of the diffusion term which generates compactly supported solutions. The key tools used in the paper are maximum-principle type arguments as well as estimates on mass concentration of solutions.

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Section: Introductionmentioning

confidence: 99%

The Patlak–Keller–Segel Model and Its Variations: Properties of Solutions via Maximum Principle

Kim¹,

Yao²

2012

SIAM J. Math. Anal.

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“…The author delivered some a priori estimates guaranteeing the boundedness of solutions (in dimension 2), which in turn justifies nonexistence of their finite or infinite time blowup (see also [24] where the results from [9] have been improved and extended to the whole space R 2 ). The similar problem with f (n) = n m (m 0) and the parameter χ = χ(n) = n q−2 (with q sufficiently small) and Ω = R d was studied recently in [25], where the authors studied the asymptotic behaviour of solutions. Further, in [26] the author proved the global existence of solutions to Eqs.…”

Section: Introductionmentioning

confidence: 99%

On the global existence of solutions to an aggregation model

Kowalczyk

Szymańska

2008

Journal of Mathematical Analysis and Applications

173

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In this paper we consider a reaction-diffusion-chemotaxis aggregation model of Keller-Segel type with a nonlinear, degenerate diffusion. Assuming that the diffusion function f (n) takes values sufficiently large, i.e. takes values greater than the values of a power function with sufficiently high power (f (n) δn p for all n > 0, where δ > 0 is a constant), we prove global-in-time existence of weak solutions. Since one of the main features of Keller-Segel type models is the possibility of blow-up of solutions in finite time, we will derive the uniform-in-time boundedness, which prevents the explosion of solutions. The uniqueness of solutions is proved provided that some higher regularity condition on solutions is known a priori. Finally, computational simulation results showing the effect of three different types of diffusion function are presented.

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“…In fact, more precise asymptotic profile for u as t → ∞ can be showing. It is proved by Luckhaus and Sugiyama [18,19] that u(t) converges to the well-known Barenblatt solution as t → ∞.…”

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confidence: 99%

Global strong solution to the semi-linear Keller–Segel system of parabolic–parabolic type with small data in scale invariant spaces

Kozono

Sugiyama

2009

Journal of Differential Equations

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In the memory of Professor Tetsuro Miyakawa MSC: 35K55We shall show existence of global strong solution to the semilinear Keller-Segel system in R n , n 3, of parabolic-parabolic type with small initial data u 0 ∈ H n r −2,r (R n ) and v 0 ∈ H n r ,r (R n ) for max{1, n/4} < r < n/2. Our method is based on the perturbation of linealization together with the L p -L q estimates of the heat semigroup and the fractional powers of the Laplace operator. As a by-product of our method, we shall prove the decay property of solutions as the time goes to infinity.In this paper, we consider the Cauchy problem of the semi-linear Keller-Segel system of parabolicparabolic type in R n for n 3,where u = u(x, t) denotes the density of amoebae and v = v(x, t) denotes the concentration of the chemo-attractant, while u 0 = u 0 (x) and v 0 = v 0 (x) are the given initial data and γ is a non-negative constant.

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Asymptotic profile with the optimal convergence rate for a parabolic equation of chemotaxis in super-critical cases

Cited by 39 publications

References 0 publications

The Patlak–Keller–Segel Model and Its Variations: Properties of Solutions via Maximum Principle

The Patlak–Keller–Segel Model and Its Variations: Properties of Solutions via Maximum Principle

On the global existence of solutions to an aggregation model

Global strong solution to the semi-linear Keller–Segel system of parabolic–parabolic type with small data in scale invariant spaces

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