Journal ArticleDOI
A Chemotaxis System with Logistic Source
J. Ignacio Tello,Michael Winkler +1 more
TLDR
In this article, the existence of global bounded classical solutions is proved under the assumption that either the space dimension does not exceed two, or that the logistic damping effect is strong enough.Abstract:
This paper deals with a nonlinear system of two partial differential equations arising in chemotaxis, involving a source term of logistic type The existence of global bounded classical solutions is proved under the assumption that either the space dimension does not exceed two, or that the logistic damping effect is strong enough Also, the existence of global weak solutions is shown under rather mild conditions Secondly, the corresponding stationary problem is studied and some regularity properties are given It is proved that in presence of certain, sufficiently strong logistic damping there is only one nonzero equilibrium, and all solutions of the non-stationary system approach this steady state for large times On the other hand, for small logistic terms some multiplicity and bifurcation results are establishedread more
Citations
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Journal ArticleDOI
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
Boundedness in the Higher-Dimensional Parabolic-Parabolic Chemotaxis System with Logistic Source
TL;DR: 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.
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
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.
Book
Navier-Stokes Equations: Theory and Numerical Analysis
TL;DR: This paper presents thediscretization of the Navier-Stokes Equations: General Stability and Convergence Theorems, and describes the development of the Curl Operator and its application to the Steady-State Naviers' Equations.
Book
Shock Waves and Reaction-Diffusion Equations
TL;DR: In this paper, the basics of hyperbolic conservation laws and the theory of systems of reaction-diffusion equations, including the generalized Morse theory as developed by Charles Conley, are presented in a way accessible to a wider audience than just mathematicians.
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.
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Initiation of slime mold aggregation viewed as an instability.
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