Journal ArticleDOI
Boundedness in the Higher-Dimensional Parabolic-Parabolic Chemotaxis System with Logistic Source
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In this article, the authors considered nonnegative solutions of the Neumann boundary value problem for the chemotaxis system in a smooth bounded convex domain, where τ > 0, χ ∈ ℝ and f is a smooth function generalizing the logistic source.Abstract:
We consider nonnegative solutions of the Neumann boundary value problem for the chemotaxis system in a smooth bounded convex domain Ω ⊂ ℝ n , n ≥ 1, where τ > 0, χ ∈ ℝ and f is a smooth function generalizing the logistic source It is shown that if μ is sufficiently large then for all sufficiently smooth initial data the problem possesses a unique global-in-time classical solution that is bounded in Ω × (0, ∞). Known results, asserting boundedness under the additional restriction n ≤ 2, are thereby extended to arbitrary space dimensions.read more
Citations
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Toward a mathematical theory of Keller–Segel models of pattern formation in biological tissues
TL;DR: In this article, a survey and critical analysis focused on a variety of chemotaxis models in biology, namely the classical Keller-Segel model and its subsequent modifications, which, in several cases, have been developed to obtain models that prevent the non-physical blow up of solutions.
Journal ArticleDOI
Global Large-Data Solutions in a Chemotaxis-(Navier–)Stokes System Modeling Cellular Swimming in Fluid Drops
TL;DR: In this paper, coupled chemotaxis (Navier and Stokes) systems generalizing the prototype have been proposed to describe the collective effects arising in bacterial suspensions in fluid drops, and they have been applied to the model of collective effects of bacterial suspensions.
Journal ArticleDOI
Blow-up in a higher-dimensional chemotaxis system despite logistic growth restriction
TL;DR: In this paper, the authors studied radially symmetric solutions of a class of chemotaxis systems generalizing the prototype { u t = Δ u − χ ∇ ⋅ ( u ∇ v ) + λ u − μ u κ, x ∈ Ω, t > 0, 0 = Δ v − m ( t ) + u, x, ∈ ǫ, t < 0, ǒ > 0, ǔ > 0.
Journal ArticleDOI
Global asymptotic stability of constant equilibria in a fully parabolic chemotaxis system with strong logistic dampening
TL;DR: In this article, it was shown that the unique nontrivial spatially homogeneous equilibrium given by u = v ≡ 1 μ is globally asymptotically stable in the sense that for any choice of suitably regular nonnegative initial data (u 0, v 0 ) such that u 0 ≢ 0, the above problem possesses a uniquely determined global classical solution ( u, v ) with ( u, v ) | t = 0 = ( u 0, v 0) which satisfies ∞ ∞ as t → ∞.
Journal ArticleDOI
Eventual smoothness and asymptotics in a three-dimensional chemotaxis system with logistic source
TL;DR: In this paper, the existence of global weak solutions to the chemotaxis system u t = Δ u − ∇ ⋅ ( u ∇ v ) + κ u − μ u 2 v t, under homogeneous Neumann boundary conditions in a smooth bounded convex domain Ω ⊂ R n, for arbitrarily small values of μ > 0.
References
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Book
Partial Differential Equations
TL;DR: In this paper, the authors present a theory for linear PDEs: Sobolev spaces Second-order elliptic equations Linear evolution equations, Hamilton-Jacobi equations and systems of conservation laws.
Book
Geometric Theory of Semilinear Parabolic Equations
TL;DR: The neighborhood of an invariant manifold near an equilibrium point is a neighborhood of nonlinear parabolic equations in physical, biological and engineering problems as mentioned in this paper, where the neighborhood of a periodic solution is defined by the invariance of the manifold.
Journal ArticleDOI
Initiation of slime mold aggregation viewed as an instability.
Evelyn Fox Keller,Lee A. Segel +1 more
TL;DR: A mathematical formulation of the general interaction of amoebae, as mediated by acrasin is presented, and a detailed analysis of the aggregation process is provided.
Journal ArticleDOI
A user’s guide to PDE models for chemotaxis
Thomas Hillen,Kevin J. Painter +1 more
TL;DR: This paper explores in detail a number of variations of the original Keller–Segel model of chemotaxis from a biological perspective, contrast their patterning properties, summarise key results on their analytical properties and classify their solution form.
Journal ArticleDOI
Aggregation vs. global diffusive behavior in the higher-dimensional Keller–Segel model
TL;DR: In this article, the authors considered the classical parabolic-parabolic Keller-Segel system with homogeneous Neumann boundary conditions in a smooth bounded domain and proved that for each q > n 2 and p > n one can find e 0 > 0 such that if the initial data ( u 0, v 0 ) satisfy L q ( Ω ) e and ∇ v 0 ‖ L p (Ω) e then the solution is global in time and bounded and asymptotically behaves like the solution of a discoupled system of linear parabolic
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Initiation of slime mold aggregation viewed as an instability.
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