arXiv subjects

Tianyuan Xu

Publications and source records attributed to Tianyuan Xu.

24 records · Page 2Linked to original sources

arXiv2019

Variational Approach of Critical Sharp Front Speeds in Density-dependent Diffusion Model with Time Delay

For the classical reaction diffusion equation, the priori speed of fronts is determined exactly in the pioneering paper (R.D. Benguria and M.C. Depassier, {\em Commun. Math. Phys.} 175:221--227, 1996) by variational characterization method. In this paper, we model the dispersal process using a density-dependent diffusion equation with time delay. We show the existence and uniqueness of sharp critical fronts, where the sharp critical front is $C^1$-smooth when the diffusion degeneracy is weaker with $1<m<2$, and the sharp critical front is non-$C^1$-smooth (piecewise smooth) when the diffusion degeneracy is stronger with $m\ge 2$. We give a new variational approach for the critical wave speed and investigate how the time delay affects the propagation mechanism of fronts. Our results provide some interesting insight into the dynamics of critical traveling wave.

math.AP↗

arXiv2018

Theoretical and numerical studies on global stability of traveling waves with oscillations for time-delayed nonlocal dispersion equations

This paper is concerned with the global stability of non-critical/critical traveling waves with oscillations for time-delayed nonlocal dispersion equations. We first theoretically prove that all traveling waves, especially the critical oscillatory traveling waves, are globally stable in a certain weighted space, where the convergence rates to the non-critical oscillatory traveling waves are time-exponential, and the convergence to the critical oscillatory traveling waves are time-algebraic. Both of the rates are optimal. The approach adopted is the weighted energy method with the fundamental solution theory for time-delayed equations. Secondly, we carry out numerical computations in different cases, which also confirm our theoretical results. Because of oscillations of the solutions and nonlocality of the equation, the numerical results obtained by the regular finite difference scheme are not stable, even worse to be blow-up. In order to overcome these obstacles, we propose a new finite difference scheme by adding artificial viscosities to both sides of the equation, and obtain the desired numerical results.

math.AP↗

arXiv2018

Propagating Profiles of a Chemotaxis Model with Degenerate Diffusion: Initial Shrinking, Eventual Smoothness and Expanding

We investigate the propagating profiles of a degenerate chemotaxis model describing the bacteria chemotaxis and consumption of oxygen by aerobic bacteria, in particular, the effect of the initial attractant distribution on bacterial clustering. We prove that the compact support of solutions may shrink if the signal concentration satisfies a special structure, and show the finite speed propagating property without assuming the special structure on attractant concentration, and obtain an explicit formula of the population spreading speed in terms of model parameters. The presented results suggest that bacterial cluster formation can be affected by chemotactic attractants and density-dependent dispersal.

math.AP↗

arXiv2017

Early and late stage profiles for a new chemotaxis model with density-dependent jump probability and quorum-sensing mechanisms

In this paper, we derive a new chemotaxis model with degenerate diffusion and density-dependent chemotactic sensitivity, and we provide a more realistic description of cell migration process for its early and late stages. Different from the existing studies focusing on the case of non-degenerate diffusion, the new model with degenerate diffusion causes us some essential difficulty on the boundedness estimates and the propagation behavior of its compact support. In the presence of logistic damping, for the early stage before tumour cells spread to the whole body, we first estimate the expanding speed of tumour region as $O(t^β)$ for $0<β<\frac{1}{2}$. Then, for the late stage of cell migration, we further prove that the asymptotic profile of the original system is just its corresponding steady state. The global convergence of the original weak solution to the steady state with exponential rate $O(e^{-ct})$ for some $c>0$ is also obtained.

math.AP↗

arXiv2014

Sandpiles and Dominos

We consider the subgroup of the abelian sandpile group of the grid graph consisting of configurations of sand that are symmetric with respect to central vertical and horizontal axes. We show that the size of this group is (i) the number of domino tilings of a corresponding weighted rectangular checkerboard; (ii) a product of special values of Chebyshev polynomials; and (iii) a double-product whose factors are sums of squares of values of trigonometric functions. We provide a new derivation of the formula due to Kasteleyn and to Temperley and Fisher for counting the number of domino tilings of a 2m x 2n rectangular checkerboard and a new way of counting the number of domino tilings of a 2m x 2n checkerboard on a Möbius strip.

math.CO↗

arXiv2010

A Note on the Critical Group of a Line Graph

This note answers a question posed by Levine in arkiv:math.CO/0906.2809. The main result is Theorem 1 which shows that under certain circumstances a critical group of a directed graph is the quotient of a critical group of its directed line graph.

math.CO↗