Effects of signal-dependent motilities in a Keller–Segel-type reaction–diffusion system

2017 ◽  
Vol 27 (09) ◽  
pp. 1645-1683 ◽  
Author(s):  
Youshan Tao ◽  
Michael Winkler
Keyword(s):  
Initial Data ◽  
Weak Solutions ◽  
Blow Up ◽  
Three Dimensional ◽  
Reaction Diffusion ◽  
Large Data ◽  
Dimensional Case ◽  
Stripe Pattern ◽  
Bounded Convex Domain ◽  
Flux Boundary Conditions

This work considers the Keller–Segel-type parabolic system [Formula: see text] in a smoothly bounded convex domain [Formula: see text], [Formula: see text], under no-flux boundary conditions, which has recently been proposed as a model for processes of stripe pattern formation via so-called “self-trapping” mechanisms. In the two-dimensional case, in stark contrast to the classical Keller–Segel model in which large-data solutions may blow up in finite time, for all suitably regular initial data the associated initial value problem is seen to possess a globally-defined bounded classical solution, provided that the motility function [Formula: see text] is uniformly positive. In the corresponding higher-dimensional setting, it is shown that certain weak solutions exist globally, where in the particular three-dimensional case this solution actually is bounded and classical if the initial data are suitably small in the norm of [Formula: see text]. Finally, if still [Formula: see text] but merely the physically interpretable quantity [Formula: see text] is appropriately small, then the above-weak solutions are proved to become eventually smooth and bounded.

To the exclusion of blow-up in a three-dimensional chemotaxis-growth model with indirect attractant production

2016 ◽  
Vol 26 (11) ◽  
pp. 2111-2128 ◽  
Author(s):  
Bingran Hu ◽  
Y. Tao
Keyword(s):  
Initial Data ◽  
Blow Up ◽  
Three Dimensional ◽  
Classical Solutions ◽  
Global Classical Solution ◽  
Infinite Time ◽  
Two Component ◽  
Flux Boundary Conditions ◽  
Growth System ◽  
Striking Contrast

This work considers the chemotaxis-growth system [Formula: see text] in a smoothly bounded domain [Formula: see text], with zero-flux boundary conditions, where [Formula: see text] and [Formula: see text] are given positive parameters. In striking contrast to the corresponding three-dimensional two-component chemo-taxis-growth system to which the global existence or blow-up of classical solutions largely remains open when [Formula: see text] is small, it is shown that whenever [Formula: see text] [Formula: see text] and [Formula: see text], for any given non-negative and suitably smooth initial data [Formula: see text] satisfying [Formula: see text], the system (⋆) admits a unique global classical solution that is uniformly-in-time bounded, which rules out the possibility of blow-up of solutions in finite time or in infinite time. Moreover, under the fully explicit condition [Formula: see text] the solution [Formula: see text] exponentially converges to the constant stationary solution [Formula: see text] in the norm of [Formula: see text] as [Formula: see text].

scholarly journals Blow-Up Criterion of Weak Solutions for the 3D Boussinesq Equations

2015 ◽  
Vol 2015 ◽  
pp. 1-6
Author(s):  
Zhaohui Dai ◽  
Xiaosong Wang ◽  
Lingrui Zhang ◽  
Wei Hou
Keyword(s):  
Weak Solutions ◽  
Blow Up ◽  
Three Dimensional ◽  
Boussinesq Equations ◽  
Interpolation Inequality ◽  
Embedding Theorem ◽  
The Earth ◽  
The Cauchy Problem ◽  
Homogeneous Besov Space ◽  
Blow Up Criterion

The Boussinesq equations describe the three-dimensional incompressible fluid moving under the gravity and the earth rotation which come from atmospheric or oceanographic turbulence where rotation and stratification play an important role. In this paper, we investigate the Cauchy problem of the three-dimensional incompressible Boussinesq equations. By commutator estimate, some interpolation inequality, and embedding theorem, we establish a blow-up criterion of weak solutions in terms of the pressurepin the homogeneous Besov spaceḂ∞,∞0.

scholarly journals Homogeneous Euler equation: blow-ups, gradient catastrophes and singularity of mappings

2021 ◽  
Author(s):  
Boris G. Konopelchenko ◽  
Giovanni Ortenzi
Keyword(s):  
Euler Equation ◽  
Euler Equations ◽  
Blow Up ◽  
Three Dimensional ◽  
Dimensional Case ◽  
Linear Combinations ◽  
First Occurrence ◽  
Different Dimensions ◽  
Characteristic Features

Abstract The paper is devoted to the analysis of the blow-ups of derivatives, gradient catastrophes and dynamics of mappings of ℝn → ℝn associated with the n-dimensional homogeneous Euler equation. Several characteristic features of the multi-dimensional case (n > 1) are described. Existence or nonexistence of blow-ups in different dimensions, boundness of certain linear combinations of blow-up derivatives and the first occurrence of the gradient catastrophe are among of them. It is shown that the potential solutions of the Euler equations exhibit blow-up derivatives in any dimenson n. Several concrete examples in two- and three-dimensional cases are analysed. Properties of ℝnu → ℝ nx mappings defined by the hodograph equations are studied, including appearance and disappearance of their singularities.

scholarly journals Formation and construction of a shock wave for 3-D compressible Euler equations with the spherical initial data

2004 ◽  
Vol 175 ◽  
pp. 125-164 ◽  
Author(s):  
Huicheng Yin
Keyword(s):  
Shock Wave ◽  
Initial Data ◽  
Euler Equations ◽  
Blow Up ◽  
Three Dimensional ◽  
Compressible Euler Equations ◽  
Lipschitz Continuous ◽  
Compressible Euler ◽  
Nondegeneracy Conditions ◽  
Weak Entropy Solution

AbstractIn this paper, the problem on formation and construction of a shock wave for three dimensional compressible Euler equations with the small perturbed spherical initial data is studied. If the given smooth initial data satisfy certain nondegeneracy conditions, then from the results in [22], we know that there exists a unique blowup point at the blowup time such that the first order derivatives of a smooth solution blow up, while the solution itself is still continuous at the blowup point. From the blowup point, we construct a weak entropy solution which is not uniformly Lipschitz continuous on two sides of a shock curve. Moreover the strength of the constructed shock is zero at the blowup point and then gradually increases. Additionally, some detailed and precise estimates on the solution are obtained in a neighbourhood of the blowup point.

Blow up for a diffusion-advection equation

1989 ◽  
Vol 113 (3-4) ◽  
pp. 181-190 ◽  
Author(s):  
Nicholas D. Alikakos ◽  
Peter W. Bates ◽  
Christopher P. Grant
Keyword(s):  
Initial Data ◽  
Boundary Conditions ◽  
Asymptotic Behaviour ◽  
Finite Time ◽  
Blow Up ◽  
Small Mass ◽  
Unit Interval ◽  
Advection Equation ◽  
Natural Question ◽  
Flux Boundary Conditions

SynopsisThese results describe the asymptotic behaviour of solutions to a certain non-linear diffusionadvection equation on the unit interval. The “no flux” boundary conditions prescribed result in mass being conserved by solutions and the existence of a mass-parametrised family of equilibria. A natural question is whether or not solutions stabilise to equilibria and if not, whether they blow up in finite time. Here it is shown that for non-linearities which characterise “fast association” there is a criticalmass such that initial data which have supercritical mass must lead to blow up in finite time. It is also shown that there exist initial data with arbitrarily small mass which also lead to blow up in finite time.

scholarly journals The blowup of solutions for 3-D axisymmetric compressible Euler equations

1999 ◽  
Vol 154 ◽  
pp. 157-169 ◽  
Author(s):  
Huicheng Yin ◽  
Qingjiu Qiu
Keyword(s):  
Asymptotic Expansion ◽  
Initial Data ◽  
Euler Equations ◽  
Blow Up ◽  
Three Dimensional ◽  
Compressible Euler Equations ◽  
Classical Solutions ◽  
Complete Asymptotic Expansion ◽  
Blowup Of Solutions ◽  
Compressible Euler

AbstractIn this paper, for three dimensional compressible Euler equations with small perturbed initial data which are axisymmetric, we prove that the classical solutions have to blow up in finite time and give a complete asymptotic expansion of lifespan.

Stabilization in a higher-dimensional attraction–repulsion chemotaxis system if repulsion dominates over attraction

2018 ◽  
Vol 28 (06) ◽  
pp. 1105-1134 ◽  
Author(s):  
Ke Lin ◽  
Chunlai Mu ◽  
Deqin Zhou
Keyword(s):  
Initial Data ◽  
Global Solutions ◽  
Large Data ◽  
Time Behavior ◽  
Higher Dimensional ◽  
All Solutions ◽  
Flux Boundary Conditions ◽  
First Results ◽  
Long Time ◽  
Chemotaxis System

The long-time behavior of solutions to an attraction–repulsion chemotaxis system is considered in this paper in a bounded domain under zero-flux boundary conditions if repulsion dominates over attraction. It is known that under the assumption that repulsion balances/overbalances attraction in two or higher dimensions, for any suitably regular initial data all solutions of this system will be global and bounded. The present work further shows that if the degradation rate of a repulsive signal is smaller than the attractive one or the repulsion is suitably strong, this global solutions converge to the steady state for arbitrarily large initial data. To the best of our knowledge, these are the first results on the large time of large-data solutions in a higher-dimensional attraction–repulsion chemotaxis system.

Asymptotic stability of spatial homogeneity in a haptotaxis model for oncolytic virotherapy

2021 ◽  
pp. 1-21
Author(s):  
Youshan Tao ◽  
Michael Winkler
Keyword(s):  
Initial Data ◽  
Blow Up ◽  
Reaction Diffusion ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Oncolytic Virotherapy ◽  
Homogeneous Distribution ◽  
Spatial Homogeneity ◽  
Image Position ◽  
Initial Boundary Value

This study considers a model for oncolytic virotherapy, as given by the reaction–diffusion–taxis system \[\begin{eqnarray*} \left\{ \begin{array}{l} u_t = \Delta u - \nabla (u\nabla v)-\rho uz, \\ v_t = - (u+w)v, \\ w_t = D_w \Delta w - w + uz, \\ z_t = D_z \Delta z - z - uz + \beta w, \end{array} \right. \end{eqnarray*}\] in a smoothly bounded domain Ω ⊂ ℝ2, with parameters D w  > 0, D z  > 0, β > 0 and ρ ⩾ 0. Previous analysis has asserted that for all reasonably regular initial data, an associated no-flux type initial-boundary value problem admits a global classical solution, and that this solution is bounded if β < 1, whereas whenever β > 1 and $({1}/{|\Omega |})\int _\Omega u(\cdot ,0) > 1/(\beta -1)$ , infinite-time blow-up occurs at least in the particular case when ρ = 0. In order to provide an appropriate complement to this, the current study reveals that for any ρ ⩾ 0 and arbitrary β > 0, at each prescribed level γ ∈ (0, 1/(β − 1)+) one can identify an L∞-neighbourhood of the homogeneous distribution (u, v, w, z) ≡ (γ, 0, 0, 0) within which all initial data lead to globally bounded solutions that stabilize towards the constant equilibrium (u∞, 0, 0, 0) with some u∞ > 0.

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