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 prevention by quadratic degradation in a higher-dimensional chemotaxis-growth model with indirect attractant production

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Jianing Xie
Keyword(s):  
Mountain Pine Beetle ◽  
Blow Up ◽  
Three Dimensional ◽  
Regularity Theory ◽  
Classical Solutions ◽  
Sobolev Regularity ◽  
Global Classical Solutions ◽  
Flux Boundary Conditions ◽  
Growth System ◽  
Energy Type

<p style='text-indent:20px;'>This paper deals with a boundary-value problem in three-dimensional smoothly bounded domains for a coupled chemotaxis-growth system generalizing the prototype</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$\begin{align} \left\{\begin{array}{ll} u_t = \Delta u-\nabla\cdot(u\nabla v)+\mu u(1-u),\quad x\in \Omega, t&gt;0,\\ { }{ v_t = \Delta v- v +w},\quad x\in \Omega, t&gt;0,\\ { }{\tau w_t+\delta w = u},\quad x\in \Omega, t&gt;0\\ \end{array}\right. \end{align} (*)$ \end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>in a smoothly bounded domain <inline-formula><tex-math id="M1">\begin{document}$ \Omega\subset\mathbb{R}^N(N\geq1) $\end{document}</tex-math></inline-formula> under zero-flux boundary conditions, which describe the spread and aggregative behavior of the Mountain Pine Beetle in forest habitat, where the parameters <inline-formula><tex-math id="M2">\begin{document}$ \mu $\end{document}</tex-math></inline-formula> as well as <inline-formula><tex-math id="M3">\begin{document}$ \delta $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ \tau $\end{document}</tex-math></inline-formula> are positive. Based on an <b>new</b> energy-type argument combined with maximal Sobolev regularity theory, it is proved that global classical solutions exist whenever</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ \mu&gt;\left\{ \begin{array}{ll} {0, \; \; \; {\rm{if}}\; \; N\leq4},\\ {\frac{(N-4)_{+}}{N-2}\max\{1,\lambda_{0}\},\; \; \; {\rm{if}}\; \; N\geq5}\\ \end{array} \right. $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>and the initial data <inline-formula><tex-math id="M5">\begin{document}$ (u_0,v_0,w_0) $\end{document}</tex-math></inline-formula> are sufficiently regular. Here <inline-formula><tex-math id="M6">\begin{document}$ \lambda_0 $\end{document}</tex-math></inline-formula> is a positive constant which is corresponding to the maximal Sobolev regularity. This extends some recent results by several authors.</p>

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.

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.

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.

Optimal decay rate of classical solutions to the compressible magnetohydrodynamic equations

2011 ◽  
Vol 141 (1) ◽  
pp. 109-126 ◽  
Author(s):  
Fucai Li ◽  
Hongjun Yu
Keyword(s):  
Decay Rate ◽  
Three Dimensional ◽  
Classical Solutions ◽  
Global Classical Solution ◽  
Small Perturbations ◽  
Magnetohydrodynamic Equation ◽  
Optimal Decay ◽  
Optimal Decay Rate ◽  
Whole Space ◽  
Constant State

The three-dimensional compressible magnetohydrodynamic equation in the whole space are studied in this paper. The global classical solution is established when the initial data are small perturbations of some given constant state. Moreover, the optimal decay rate of the solution is also obtained.

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.

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