arXiv subjects

Shen Bian

Publications and source records attributed to Shen Bian.

16 recordsLinked to original sources

arXiv2026

On a two-species Keller-Segel model with degenerate diffusion and two stimuli

This paper investigates a two-species chemotaxis system with degenerate diffusion in the whole space $\R^d(d \ge 3)$. The diffusion of each species is governed by porous-medium-type operators. Under suitable conditions on the diffusion and aggregation exponents, we establish the global existence of uniformly bounded weak solutions. The proof hinges on a refined energy estimate that exploits the regularizing effect of degenerate diffusion to counteract the chemotactic aggregation. Furthermore, under additional assumptions on the system parameters, we derive exponential convergence rates of the solutions toward the constant steady state. Our results reveal that sufficiently strong degenerate diffusion ensures global boundedness and exponential stabilization in the whole space.

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arXiv2026

Sharp threshold for a one-dimensional thin film equation in the supercritical case

We study a one-dimensional thin film equation combining competitive effects of aggregation and repulsion, where repulsion is modeled by fourth-order diffusion and aggregation by backward second-order degenerate diffusion with exponent $m>0$. Under natural regularity constraints, we prove that for every $m>0$, there exists a unique (up to the mass-critical case $m=3$) nonnegative, radially decreasing steady state $U_*$ which coincides with the extremal function of the sharp Sz.-Nagy inequality and is simultaneously the global minimizer of the free energy. Using this variational characterization in the supercritical regime $3<m<\infty$, we show that finite-time blow-up occurs for all initial data whose initial free energy lies below the positive threshold $F(U_*)$, provided the $L^{m+1}$-norm of the initial datum exceeds that of $U_*$. Conversely, if the $L^{m+1}$-norm is below that of $U_*$, the solution exists globally and its second moment diverges as $t\to\infty$. This sharp criterion significantly extends the previously known blow-up condition requiring negative free energy to a much wider class of initial data (see \cite{BP00}). Our results identify the steady state $U_*$ as the critical pivot linking variational structure to dynamical behavior, and provide a constructive method to determine blow-up versus global existence via an explicit $L^{m+1}$-norm comparison.

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arXiv2026

Steady states and dynamics of a higher dimensional thin film equation

We study a higher-dimensional thin film equation that incorporates competitive effects between aggregation and repulsion, where repulsion is modeled by fourth-order diffusion and aggregation by backward second-order degenerate diffusion, with a degenerate diffusion exponent $m>0$. We first conduct a systematic analysis of the existence and geometric properties of steady-state solutions for all $m>0$, revealing a critical threshold $m^*=(d+2)/(d-2)$ for variational compactness and solution structure. For $0 < m < m^*$, we then prove that, under natural regularity constraints, radially decreasing steady states coincide with both the extremals of the Gagliardo-Nirenberg-Sobolev inequality and the global minimizers of the free energy. Moreover, we establish the uniqueness of such steady states for $m \neq 1 + 2/d$. Furthermore, in the supercritical regime $1 + 2/d < m < m^*$, we identify a sharp threshold given by the $L^{m+1}$ norm of the unique radial steady-state solution, which distinguishes between global existence for initial data below the threshold and finite-time blow-up for initial data above the threshold. The main contribution of this work is to use steady-state solutions as a theoretical pivot to construct a unified analytical framework that connects parameter classification, variational structure and dynamic behavior. This framework elucidates how the regularity barrier prevents infinite energy descent and selects stable equilibrium states, and thus predicts the global evolution of the system, thereby providing a unified variational principle for understanding steady-state selection and dynamic bifurcations in such higher-order degenerate diffusion equations.

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arXiv2025

Critical threshold for a two-species chemotaxis system with the energy critical exponent

We consider a two-species chemotaxis model in $\R^d(d \ge 3)$ featuring nonlinear porous medium-type diffusion and nonlocal attractive power-law interaction. Here, the nonlinear diffusion is chosen to be $1/m_1+1/m_2=(d+2)/d$ in such a way that the associated free energy is conformal invariant, and there are radially symmetric, non-increasing and non-compactly supported stationary solutions $(U_s(x),V_s(x))$. We analyze the conditions on initial data $(u_0,v_0)$ under which attractive forces dominate over diffusion, and further classify the global existence and finite time blow-up of dynamical solutions by virtue of these stationary solutions. Specifically, the solution $(u,v)(x,t)$ exists globally in time if the initial data satisfy $\|u_0\|_{L^{m_1}(\R^d)}<\|U_s\|_{L^{m_1}(\R^d)}$ and $\|v_0\|_{L^{m_2}(\R^d)}<\|V_s\|_{L^{m_2}(\R^d)}$. In contrast, there are blowing-up solutions when $\|u_0\|_{L^{m_1}(\R^d)}>\|U_s\|_{L^{m_1}(\R^d)}$ and $\|v_0\|_{L^{m_2}(\R^d)}>\|V_s\|_{L^{m_2}(\R^d)}$.

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arXiv2025

Adaptive dynamics of nonlocal competition models with heterogeneous resources

We investigate the long-time behavior of phenotype-structured models describing evolutionary dynamics of asexual populations, and analyze the joint effects of nonlocal interactions and spatial resource distributions on the global dynamics of the two species. In the first part, we consider an integro-differential system without diffusion terms, where phenotypic changes are absent and the spatial distribution of resources for one species is heterogeneous while that of the other is homogeneous. Using an entropy method to address nonlocal interactions and resource heterogeneity, we prove that the species subject to heterogeneous resources converges to a Dirac mass concentrated at the peak of the phenotypic fitness landscape, which establishes the selection of the best adapted trait. Numerical experiments further provide a sufficient criterion to identify the positions of fitness peaks. In the second part, we extend our study to a nonlocal reaction-diffusion system involving a linear diffusion operator, where heritable phenotypic changes occur and both resources are spatially heterogeneous. Through an appropriate transformation, we overcome the challenges induced by resource heterogeneity and prove that the long-time limits of the two species under different interspecific competitive coefficients are given by distinct steady states of the parabolic system, with concentrations at the maxima of their respective resource functions. Numerical results confirm the predictions and further reveal phenomena beyond the theoretical analysis.

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arXiv2024

The Keller-Segel model with mass critical exponent

We consider a Keller-Segel model with non-linear porous medium type diffusion and non-local attractive power law interaction, focusing on potentials that are less singular than Newtonian interaction. Here, the nonlinear diffusion is chosen to be $m=2-\frac{2s}{d}$, in which case the steady states are compactly supported. We analyse under what conditions on the initial data the regime that attractive forces are stronger than diffusion occurs and classify the conditions for global existence and finite time blow-up of solutions. It is shown that there exists a threshold value which is characterized by the optimal constant of a variant of the Hardy-Littlewood-Sobolev inequality. Specifically, the solution will exist globally if the initial data is below the threshold, while the solution blows up in finite time when the initial data is above the threshold.

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arXiv2024

On selection dynamics for a nonlocal phenotype-structured model

This paper is devoted to the analysis of the long-time behavior of a phenotypic-structured model where phenotypic changes do not occur. We give a mathematical description of the process in which the best adapted trait is selected in a given environment created by the total population. It is exhibited that the long-time limit of the unique solution to the nonlocal equation is given by a Dirac mass centered at the peak of the fitness within or at the boundary of the region where the initial data is positive. Specially, If the peak of the fitness can't be in the support of the solution, then the infinite time blow-up of the solution occurs near the boundary of the region where the solution is positive. Moreover, our numerical results facilitate a deeper understanding of identifying the position of the centers.

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arXiv2023

The aggregation-diffusion equation with the intermediate exponent

We consider a Keller-Segel model with non-linear porous medium type diffusion and nonlocal attractive power law interaction, focusing on potentials that are less singular than Newtonian interaction. Here, the nonlinear diffusion is chosen to be $\frac{2d}{d+2s}<m<2-\frac{2s}{d}$ in which case the steady states are compactly supported. We analyse under which conditions on the initial data the regime that attractive forces are stronger than diffusion occurs and classify the global existence and finite time blow-up of solutions. It is shown that there is a threshold value which is characterized by the optimal constant of a variant of Hardy-Littlewood-Sobolev inequality such that the solution will exist globally if the initial data is below the threshold, while the solution blows up in finite time when the initial data is above the threshold.

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arXiv2023

The aggregation-diffusion equation with energy critical exponent

We consider a Keller-Segel model with non-linear porous medium type diffusion and nonlocal attractive power law interaction, focusing on potentials that are less singular than Newtonian interaction. Here, the nonlinear diffusion is chosen to be $m=\frac{2d}{d+2s}$ in such a way that the associated free energy is conformal invariant and there is a family of stationary solutions $U(x)=c\left(\fracλ{λ^2+|x-x_0|^2}\right)^{\frac{d+2s}{2}}$ for any constant $c$ and some $λ>0, x_0 \in \R^d.$ We analyze under which conditions on the initial data the regime that attractive forces are stronger than diffusion occurs and classify the global existence and finite time blow-up of dynamical solutions by virtue of stationary solutions. Precisely, solutions exist globally in time if the $L^m$ norm of the initial data $\|u_0\|_{L^m(\R^d)}$ is less than the $L^m$ norm of stationary solutions $\|U(x)\|_{L^m(\R^d)}$. Whereas there are blowing-up solutions for $\|u_0\|_{L^m(\R^d)}>\|U(x)\|_{L^m(\R^d)}$.

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arXiv2022

Keller-Segel model with Logarithmic Interaction and nonlocal reaction term

We investigate the global existence and blow-up of solutions to the Keller-Segel model with nonlocal reaction term $u\left(M_0-\int_{\R^2} u dx\right)$ in dimension two. By introducing a transformation in terms of the total mass of the populations to deal with the lack of mass conservation, we exhibit that the qualitative behavior of solutions is decided by a critical value $8π$ for the growth parameter $M_0$ and the initial mass $m_0$. For general solutions, if both $m_0$ and $M_0$ are less than $8π$, solutions exist globally in time using the energy inequality, whereas there are finite time blow-up solutions for $M_0>8π$ (It involves the case $m_0<8π$) with any initial data and $M_0<8π 0$, then all the radially symmetric solutions are vanishing in $L_{loc}^1(\R^2)$ as $t \to \infty$. If the initial data $u_0(r)>\frac{m_0}{M_0} \frac{8 λ}{(r^2+λ)^2}$ for some $λ>0$, then there could exist a radially symmetric solution satisfying a mass concentration at the origin as $t \to \infty.$

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arXiv2021

On the cauchy problem with degenerate diffusion and nonlocal nonlinear sources

This paper is devoted to the analysis of non-negative solutions for a generalisation of the parabolic equation with porous medium like nonlinear diffusion and nonlinear nonlocal reaction. We investigate under which conditions equilibration between two competing effects, repulsion modelled by nonlinear diffusion and aggregation modelled by nonlinear reaction, occurs. Precisely, we exhibit that the qualitative behavior of solutions is decided by the nonlinear diffusion which is chosen in such a way that its scaling and the reaction term coincide, i.e. that there is a critical exponent $m+2/n$ for the reaction exponent $α,$ solutions exist globally with uniformly upper bounds in the case of (i)$1\le α m+2/n$ for small initial data and (iii) $α=m+2/n$ for small mass capacity $M_0$. In the case of (ii) and (iii), the decay properties of the solution are also discussed. Moreover, numerical simulations are carried out to verify the theoretical analysis and explore other issues that lie beyond the scope of the analysis.

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arXiv2019

Global existence in the critical and subcritical cases to the Fisher-KPP model with nonlocal nonlinear reaction

The Cauchy problem considered in this paper is the following \begin{align} \left\{ \begin{array}{ll} u_t=Δu+u^α\left(M_0- \int_{\mathbb{R}^n} u(x,t)dx\right),\quad & x \in \mathbb{R}^n, t>0, u(x,0)=U_0(x)\geq 0,\quad & x \in \mathbb{R}^n. \end{array} \right. \end{align} where $M_0>0, α>1, n \ge 3$. When the coefficient $M_0-\int_{\mathbb{R}^n} u(x,t) dx$ remains positive, \er{nkpp0} is analogous to \begin{align} \left\{ \begin{array}{ll} u_t=Δu+u^α,\quad & x \in \mathbb{R}^n, t>0, u(x,0)=U_0(x)\geq 0,\quad & x \in \mathbb{R}^n. \end{array} \right. \end{align} It is well known that when $1<α\le 1+2/n$, the local solution of \er{fujita} blows up in finite time as long as the initial value is nontrivial. The present paper forms a contrast to \er{fujita} and shows the global existence of solutions to \er{nkpp0} for $1<α\le 1+2/n$ by dealing with the mathematical challenge which is from the nonlocal term $\int_{\mathbb{R}^n} u dx$. It's proved that when $1<α<1+2/n$, such a global bound is obtained for any positive $M_0$ and any non-negative initial data. While if $α=1+2/n$, then the global solution does exist for sufficiently small $M_0$ and any non-negative initial data. Furthermore, the large time behavior of the global solution is also discussed for $α=1+2/n$. Besides, this paper establishes the hyper-contractivity of a global solution in $L^\infty(\mathbb{R}^n)$ with $U_0 \in L^1(\mathbb{R}^n)$ for the case $α=1+2/n$.

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arXiv2018

Nonlocal nonlinear reaction preventing blow-up in Keller-Segel system

This paper is devoted to the analysis of non-negative solutions for the chemotaxis model with nonlocal source in bounded domain. The qualitative behavior of solutions is determined by the nonlinearity from the aggregation and the reaction. The nonlocal nonlinear Fisher-KPP reaction helps preventing blow-up phenomena in chemotaxis system. For appropriately chosen exponents, the global existence of classical solutions is proved with arbitrary initial data, where a modified Moser-Alikakos iteration method plays a key role in the a priori estimates.

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arXiv2017

Chemotaxis model with subcritical exponent in nonlocal reaction

This paper deals with a parabolic-elliptic chemotaxis system with nonlocal type of source in the whole space. It's proved that the initial value problem possesses a unique global solution which is uniformly bounded. Here we identify the exponents regimes of nonlinear reaction and aggregation in such a way that their scaling and the diffusion term coincide (see Introduction). Comparing to the classical KS model (without the source term), it's shown that how energy estimates give natural conditions on the nonlinearities implying the absence of blow-up for the solution without any restriction on the initial data.

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arXiv2015

A nonlocal reaction diffusion equation and its relation with Fujita exponent

This paper is concerned with a type of nonlinear reaction-diffusion equation, which arises from the population dynamics. The equation includes a certain type reaction term $u^α(1- σ\int_{\R^n}u^βdx)$ of dimension $n \ge 1$ and $σ>0$. An energy-methods-based proof on the existence of global solutions is presented and the qualitative behavior of solution which is decided by the choice of $α,β$ is exhibited. More precisely, for $1 \le α<1+(1-2/p)β$, where $p$ is the exponent appears in Sobolev's embedding theorem defined in \er{p}, the equation admits a unique global solution for any nonnegative initial data. Especially, in the case of $n\geq 2$ and $β=1$, the exponent $α<1+2/n$ is exactly the well-known Fujita exponent. The global existence result obtained in this paper shows that by switching on the nonlocal effect, i.e., from $σ=0$ to $σ>0$, the solution's behavior differs distinctly, that's, from finite time blow-up to global existence.

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arXiv2015

Global existence and asymptotic behavior of solutions to a nonlocal Fisher-KPP type problem

In this work, we consider a nonlocal Fisher-KPP reaction-diffusion problem with Neumann boundary condition and nonnegative initial data in a bounded domain in $\mathbb{R}^n (n \ge 1)$, with reaction term $u^α(1-m(t))$, where $m(t)$ is the total mass at time $t$. When $α\ge 1$ and the initial mass is greater than or equal to one, the problem has a unique nonnegative classical solution. While if the initial mass is less than one, then the problem admits a unique global solution for $n=1,2$ with any $1 \le α<2$ or $n \ge 3$ with any $1 \le α< 1+2/n$. Moreover, the asymptotic convergence to the solution of the heat equation is proved. Finally, some numerical simulations in dimensions $n=1,2$ are exhibited. Especially, for $α>2$ and the initial mass is less than one, our numerical results show that the solution exists globally in time and the mass tends to one as time goes to infinity.

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