arXiv subjects

Chunhua Jin

Publications and source records attributed to Chunhua Jin.

8 recordsLinked to original sources

arXiv2026

Classification of self-similar singular solutions with large mass for Keller-Segel model with signal consumption

In this paper, we concentrate on investigating the self-similar singular solutions of Keller-Segel model with signal consumption ($-uv^α$) and singular sensitivity. We perform a detailed exploration into the existence and decay rate of self-similar solutions, particularly, the permissibility of arbitrary mass for these solutions across all possible cases. Based on these findings, we can delve deeper into verifying that these self-similar solutions $(u, v)$ exhibit varying degrees of singularity depending on the value of $α$ and the spatial dimension. Our analysis reveals that the component $u$ (with arbitrary mass) of the solution consistently behaves analogous to heat kernel, that is, $u$ exhibiting a Dirac $δ$ initial singularity identical to that of the fundamental solution, and converges to $0$ in the sense of the $L^p$-norm ($p>1$) as time approaches infinity. However, the initial behavior of the other component $v$ varies significantly based on the value of $α$ and the spatial dimension, exhibiting regularity (not singular), less singularity, or strong singularity (more singular than fundamental solution). Moreover, both $u$ and $v$ undergo instantaneous smoothing, becoming smooth immediately after $t>0$. This phenomenon reveals the adaptive strategies of cells in high-density aggregation environments to prevent resource depletion, reflecting an optimization process of self-organizing behavior.

math.AP↗

arXiv2026

Self-similar Dynamics in the Critical $p$-Laplacian Patlak-Keller-Segel Model: Shrinking Blow-up and Expanding Propagation

In this paper, we study the following Patlak-Keller-Segel model with $p$-Laplacian diffusion \begin{align*} \left\{ \begin{aligned} &ρ_t=\nabla \cdot \left( \left| \nabla ρ\right|^{p-2}\nabla ρ\right) -χ\nabla \cdot \left( ρ\nabla c \right), &0=\varDelta c+ρ^m, \end{aligned}\right. \end{align*} and the exponent $m>0$ is chosen as $$ m = \frac{(p-2)N + p}{N}. $$ This relation ensures the scale invariance of the system and is conjectured to be the critical exponent that separates global boundedness from finite-time blow-up. We prove that, at the critical threshold $m=\frac{(p-2)N + p}{N}$, the system indeed admits finite-time blow-up solutions. More precisely, in the slow diffusion regime $p>2$, there exist backward self-similar blow-up solutions that are radially decreasing, compactly supported, and concentrate into a Dirac $δ$-measure at the blow-up time $T$; and their supports shrink toward the origin at the rate $(T-t)^{\frac1{mN}}$. For the fast diffusion case $1<p\le 2$, we show that there are no backward self-similar blow-up solutions with finite-mass. Additionally, we also explore forward self-similar solutions in both the slow diffusion and fast diffusion cases. These solutions also carry finite mass and exhibit a Dirac $δ$-singularity at the initial moment. Specifically, in the slow diffusion case, the support expands at the rate $t^{\frac1{mN}}$, whereas in the fast diffusion case, the solution becomes strictly positive for all positive times. Our work provides the first blow up analysis for the $p$-Laplacian Keller-Segel system when $p\ne 2$, and it confirms that the exponent $m$ given above is indeed the sharp threshold between global existence and finite time singularity formation.

math.AP↗

arXiv2025

Long time dynamics of the Cauchy problem for the predator-prey model with cross-diffusion

This paper is concerned with a predator-prey model in $N$-dimensional spaces ($N=1, 2, 3$), given by \begin{align*}\left\{\begin{aligned} &\frac{\partial u}{\partial t}=Δu-χ\nabla\cdot(u\nabla v),\\ &\frac{\partial v}{\partial t}=Δv+ξ\nabla\cdot(v\nabla u), \end{aligned}\right. \end{align*} which describes random movement of both predator and prey species, as well as the spatial dynamics involving predators pursuing prey and prey attempting to evade predators. It is shown that any global strong solutions of the corresponding Cauchy problem converge to zero in the sense of $L^p$-norm for any $1<p\le \infty$, and also converge to the heat kernel with respect to $L^p$-norm for any $1\le p\le \infty$. In particular, the decay rate thereof is optimal in the sense that it is consistent with that of the heat equation in $\mathbb R^N$ ($N=2, 3$). Undoubtedly, the global existence of solutions appears to be among the most challenging topic in the analysis of this model. Indeed even in the one-dimensional setting, only global weak solutions in a bounded domain have been successfully constructed by far. Nevertheless, to provide a comprehensive understanding of the main results, we append the conclusion on the global existence and asymptotic behavior of strong solutions, although certain smallness conditions on the initial data are required.

math.AP↗

arXiv2021

Bounded weak and strong time periodic solutions to a three-dimensional chemotaxis-Stokes model with porous medium diffusion

In this paper, we study the time periodic problem to a three-dimensional chemotaxis-Stokes model with porous medium diffusion $Δn^m$ and inhomogeneous mixed boundary conditions. By using a double-level approximation method and some iterative techniques, we obtain the existence and time-space uniform boundedness of weak time periodic solutions for any $m>1$. Moreover, we improve the regularity for $m\le\frac{4}{3}$ and show that the obtained periodic solutions are in fact strong periodic solutions.

math.AP↗

arXiv2021

Global bounded solution in three-dimensional chemotaxis-Stokes model with arbitrary porous medium slow diffusion

In this paper, we study the consumption-chemotaxis-Stokes model with porous medium slow diffusion in a three dimensional bounded domain with zero-flux boundary conditions and no-slip boundary condition. In recent ten years, many efforts have been made to find the global bounded solutions of chemotaxis-Stokes system in three dimensional space. Although some important progress has been carried out in some papers, as mentioned by some authors, the question of identifying an optimal condition on m ensuring global boundedness in the three-dimensional framework remains an open challenge. In the present paper, we put forward a new estimation technique, completely proved the existence of global bounded solutions for arbitrary slow diffusion case, and partially answered the open problem proposed by Winkler.

math.AP↗

arXiv2018

Two-level Attention with Two-stage Multi-task Learning for Facial Emotion Recognition

Compared with facial emotion recognition on categorical model, the dimensional emotion recognition can describe numerous emotions of the real world more accurately. Most prior works of dimensional emotion estimation only considered laboratory data and used video, speech or other multi-modal features. The effect of these methods applied on static images in the real world is unknown. In this paper, a two-level attention with two-stage multi-task learning (2Att-2Mt) framework is proposed for facial emotion estimation on only static images. Firstly, the features of corresponding region(position-level features) are extracted and enhanced automatically by first-level attention mechanism. In the following, we utilize Bi-directional Recurrent Neural Network(Bi-RNN) with self-attention(second-level attention) to make full use of the relationship features of different layers(layer-level features) adaptively. Owing to the inherent complexity of dimensional emotion recognition, we propose a two-stage multi-task learning structure to exploited categorical representations to ameliorate the dimensional representations and estimate valence and arousal simultaneously in view of the correlation of the two targets. The quantitative results conducted on AffectNet dataset show significant advancement on Concordance Correlation Coefficient(CCC) and Root Mean Square Error(RMSE), illustrating the superiority of the proposed framework. Besides, extensive comparative experiments have also fully demonstrated the effectiveness of different components.

cs.CV↗

arXiv2018

Global solvability and stability to a nutrient-taxis model with porous medium slow diffusion

In this paper, we study a nutrient-taxis model with porous medium slow diffusion \begin{align*} \left\{ \begin{aligned} &u_t=Δu^m-χ\nabla\cdot(u\nabla v)+ξuv-ρu, \\ &v_t-Δv=-vu+μv(1-v), \end{aligned}\right. \end{align*} in a bounded domain $Ω\subset \mathbb R^3$ with zero-flux boundary condition. It is shown that for any $m>\frac{11}4-\sqrt 3$,the problem admits a global weak solution for any large initial datum. We divide the study into three cases,(i) $ξμ=0, ρ\ge 0$; (ii) $ξμρ>0$; (iii) $ξμ>0$, $ρ=0$. In particular, for Case (i) and Case (ii), the global solutions are uniformly bounded. Subsequently, the large time behavior of these global bounded solutions are also discussed. At last, we also extend the results to the coupled chemotaxis-Stokes system. Important progresses for chemotaxis-Stokes system with $m>\frac 76$, $m>\frac 87$ and $m>\frac 98$ have been carried out respectively by \cite{W2, TW2, W3}, but leave a gap for $1<m\le \frac98$. Our result for chemotaxis-Stokes system supplements part of the gap $(\frac{11}4-\sqrt 3, \frac 98)$. Here $\frac{11}4-\sqrt 3\approx 1.018$.

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↗