Journal of Partial Differential Equations
type: channel
description: Journal of Partial Differential Equations (JPDE) publishes high quality papers and short communications in theory, applications and numerical analysis of partial differential equations. Papers addressing new theoretical techniques, novel ideas, and new analysis tools are suitable topics for the journal. JPDE is a peer-reviewed international journal which is founded by Peking University, Institute of Applied Physics and Computational Mathematics, and Zhengzhou University, with an editorial office based in Zhengzhou, China.
publisher: Global Science Press
language: en
publication name: Journal of Partial Differential Equations
issn: 1000-940X
items: Free Transport Equation and Hyperbolic Schrödinger Equation via Wigner Transformation
items: Regularity Criteria for 3D Liquid Crystal Flows in Besov Space
items: Spectral Properties of the Sub-Laplacian on 2-Step Stratified Lie Groups Without the Moore-Wolf Condition
items: On Concentration of Real Solutions for Fractional Helmholtz Equation
items: Monotonic Behavior of Positive Solutions for Semi-Linear Parabolic Equations with Uniformly Elliptic Non-Local Operators in Half-Space
items: Eigenvalue Problem for a Class of Nonlinear Operators Containing $p$(·)-Laplacian in a Variable Exponent Sobolev Space
items: Well-Posedness of 5th-Order KdV Equation Posed on a Finite Domain with Nonlinear Boundary Values
items: The Existence and Exact Multiplicity of One-Sign Solutions for Semilinear Elliptic Problems in $\mathbb{R}^N$
items: Several Parameter Type Logarithmic Sobolev Inequalities
items: Multiplicity of Solutions for a Class of Critical Choquard Equation in a Bounded Domain
items: Existence of Multiple Weak Solutions for Nonlinear Problems on the Sierpiński Gasket
items: Multiple Periodic Solutions of Allen-Cahn System Involving Fractional Laplacian
items: Uniform Attractors for Non-Autonomous Reaction-Diffusion Equation with Mixed Delay
items: A Food Chain Model with a Protection Zone
items: Fine Behaviors of Eigenvalues and Eigenfunctions of the Lane-Emden Problem in Dimension Two
items: Global Large Solution for the Tropical Climate Model Without Thermal Diffusion
items: Global and Non-Global Solutions for Pseudo-Parabolic Equation with Singular Potential
items: Boundary Regularity for an Even Order Elliptic System in the Critical Dimension
items: Asymptotic Behavior in a Quasilinear Two-Species Chemotaxis System with Nonlinear Sensitivity and Nonlinear Signal Secretion
items: Estimates on Weighted $L^q$-Norms of the Nonstationary 3D Navier-Stokes Flow in an Exterior Domain
items: Parabolic System Related to the P-Laplician with Degeneracy on the Boundary
items: Structural Stability of p(x)-Laplace Problems with Fourier Type Boundary Condition
items: The Liouville Type Theorem for a System of Nonlinear Integral Equations on Exterior Domain
items: A Second-order Hyperbolic Chemotaxis Model
items: Bifurcation and Stability Analysis in Complex Cross-Diffusion Mathematical Model of Phytoplankton-Fish Dynamics
items: Transportation Inequalities for Multivalued Stochastic Evolution Equations
items: Mixed Layer Problem of a Three-Dimensional Drift-Diffusion Model for Semiconductors
items: Solutions to a 3D Burgers Equation with Initial Discontinuity That Are Two Disjoint Spheres
items: Entropy Solution to Nonlinear Elliptic Problem with Non-local Boundary Conditions and L¹-data
items: Semi-linear Elliptic Equations on Graph

Free Transport Equation and Hyperbolic Schrödinger Equation via Wigner Transformation
type: item
description: In this work, we study the regularity of the Cauchy problem for the free transport equation, and by using the inverse Wigner transformation, we reduce this problem to the Cauchy problem of a class of linear homogeneous hyperbolic Schrödinger equation. We prove firstly the analytical smoothing effect of Cauchy problem for Schrödinger type equation if the initial datum is exponential decay. Finally we prove the directional propagation of the exponential decay and also analytic regularity for free transport equation.
subject: Articles
  topic: section
subject: Free ransport equation
  topic: keywords
subject: hyperbolic Schrödinger equation
  topic: keywords
subject: analytical smoothing effect
  topic: keywords
subject: exponential decay
  topic: keywords
creator: Liutao Guo
creator: Chaojiang Xu
rights: Copyright (c) 2026 Journal of Partial Differential Equations https://creativecommons.org/licenses/by-nc-nd/4.0
date: 2026-03-07
publication date: 2026-03-07
starting page: 107
ending page: 125
doi: 10.4208/jpde.v39.n1.6

Regularity Criteria for 3D Liquid Crystal Flows in Besov Space
type: item
description: This note is devoted to investigating regularity criteria for 3D liquid crystal flows in Besov space.
subject: Articles
  topic: section
subject: Regularity criteria
  topic: keywords
subject: Liquid crystal flows
  topic: keywords
subject: Navier-Stokes equations
  topic: keywords
subject: Besov space
  topic: keywords
creator: Junting Feng
creator: Helin Guo
creator: Lingling Zhao
rights: Copyright (c) 2026 Journal of Partial Differential Equations https://creativecommons.org/licenses/by-nc-nd/4.0
date: 2026-03-07
publication date: 2026-03-07
starting page: 91
ending page: 106
doi: 10.4208/jpde.v39.n1.5

Spectral Properties of the Sub-Laplacian on 2-Step Stratified Lie Groups Without the Moore-Wolf Condition
type: item
description: In this paper, we consider the structure of the $L^2$-spectrum of the sub-Laplacian on 2-step stratified Lie groups by using the theory of unitary irreducible representations and Hermit functions. We extend the results for Heisenberg group and H-type Lie group to more general 2-step stratified Lie groups without the Moore-Wolf condition.
subject: Articles
  topic: section
subject: Sub-Laplacian
  topic: keywords
subject: spectrum
  topic: keywords
subject: stratified Lie groups
  topic: keywords
creator: Zhipeng Yang
rights: Copyright (c) 2026 Journal of Partial Differential Equations https://creativecommons.org/licenses/by-nc-nd/4.0
date: 2026-03-07
publication date: 2026-03-07
starting page: 70
ending page: 90
doi: 10.4208/jpde.v39.n1.4

On Concentration of Real Solutions for Fractional Helmholtz Equation
type: item
description: This paper studies the nonlinear fractional Helmholtz equation $(-\Delta)^su-k^2u=Q(x)|u|^{p-2}u,\quad\text{in }\mathbb{R}^N,\,N\geq3,\tag{0.1}$ where $\frac{N}{N+1}<s<\frac{N}{2},\quad\frac{2(N+1)}{N-1}<p<\frac{2N}{N-2s}$ are two real exponents, and the coefficient $Q(x)$ is a bounded continuous, nonnegative function that satisfies the condition $\limsup\limits_{|x|\to\infty} Q(x) < \sup\limits_{x\in\mathbb{R}^N} Q(x). \tag{0.2}$ For sufficiently large $k>0$, the existence of real-valued solutions to $(0.1)$ is established. Furthermore, as $k\rightarrow∞$, it is shown that the sequence of solutions associated with the ground states of a dual equation concentrates, after rescaling, at global maximum points of the function $Q(x).$
subject: Articles
  topic: section
subject: Fractional Helmholtz equation
  topic: keywords
subject: concentration of solutions
  topic: keywords
subject: dual variational method
  topic: keywords
subject: Lusternik-Schnirelman category
  topic: keywords
creator: Zifei Shen
creator: Shuijin Zhang
rights: Copyright (c) 2026 Journal of Partial Differential Equations https://creativecommons.org/licenses/by-nc-nd/4.0
date: 2026-03-07
publication date: 2026-03-07
starting page: 51
ending page: 69
doi: 10.4208/jpde.v39.n1.3

Monotonic Behavior of Positive Solutions for Semi-Linear Parabolic Equations with Uniformly Elliptic Non-Local Operators in Half-Space
type: item
description: We address the problem given by the following partial differential equation: some semi-Linear parabolic equations with uniformly elliptic non-local operators in Half-Space. Initially, we establish a generalized weighted average inequality and a maximum principle in unbounded domains, which are crucial for the sliding method. Then, we employ sliding to demonstrate the monotonicity of bounded positive solutions. In this paper, we will remove the monotonicity assumption of the kernel function $a(x)$ by using the sliding method. The techniques employed in the process of this method have applications to other problems related to uniformly elliptic operators.
subject: Articles
  topic: section
subject: Fractional parabolic problem
  topic: keywords
subject: uniform elliptic nonlocal operator
  topic: keywords
subject: monotonicity
  topic: keywords
subject: sliding method
  topic: keywords
creator: Qing Guo
creator: Yuhang Zhang
rights: Copyright (c) 2026 Journal of Partial Differential Equations https://creativecommons.org/licenses/by-nc-nd/4.0
date: 2026-03-07
publication date: 2026-03-07
starting page: 31
ending page: 50
doi: 10.4208/jpde.v39.n1.2

Eigenvalue Problem for a Class of Nonlinear Operators Containing $p$(·)-Laplacian in a Variable Exponent Sobolev Space
type: item
description: In this paper, we consider an eigenvalue problem for a class of nonlinear operators containing $p(·)$-Laplacian and mean curvature operator with mixed boundary conditions. More precisely, we are concerned with the problem with the Dirichlet condition on a part of the boundary and the Steklov boundary condition on an another part of the boundary. We show that the eigenvalue problem has infinitely many eigenpairs by using the celebrated Ljusternik-Schnirelmann principle of the calculus of variation. Moreover, in a variable exponent Sobolev space, there are two cases where the infimum of all eigenvalues is equal to zero and is positive.
subject: Articles
  topic: section
subject: Eigenvalue problem
  topic: keywords
subject: $p(·)$-Laplacian
  topic: keywords
subject: mean curvature operator
  topic: keywords
subject: mixed boundary value problem
  topic: keywords
subject: variable exponent Sobolev space
  topic: keywords
creator: Junichi Aramaki
rights: Copyright (c) 2026 Journal of Partial Differential Equations https://creativecommons.org/licenses/by-nc-nd/4.0
date: 2026-03-07
publication date: 2026-03-07
starting page: 1
ending page: 30
doi: 10.4208/jpde.v39.n1.1

Well-Posedness of 5th-Order KdV Equation Posed on a Finite Domain with Nonlinear Boundary Values
type: item
description: In this article, we investigate the well-posedness of the initial-boundary value problem (I-B-V problem) for the fifth-order KdV equation posed on a finite domain with nonlinear boundary conditions. Firstly, we establish various a priori estimates, including Kato smoothing effects, sharp trace regularity, and nonlinear estimates. Subsequently, we demonstrate that the initial-boundary value problem of the fifth-order KdV equation with quadratic boundary feedbacks is locally well-posed for the appropriately chosen initial value and boundary values.
subject: Articles
  topic: section
subject: 5th-order KdV equation
  topic: keywords
subject: I-B-V problem
  topic: keywords
subject: quadratic boundary feedbacks
  topic: keywords
subject: a prior estimates
  topic: keywords
subject: local well-posedness
  topic: keywords
creator: Wanmei Hou
creator: Xiangqing Zhao
rights: Copyright (c) 2025 Journal of Partial Differential Equations https://creativecommons.org/licenses/by-nc-nd/4.0
date: 2025-12-27
publication date: 2025-12-27
starting page: 494
ending page: 503
doi: 10.4208/jpde.v38.n4.7

The Existence and Exact Multiplicity of One-Sign Solutions for Semilinear Elliptic Problems in $\mathbb{R}^N$
type: item
description: In this work, we study the existence of one-sign solutions for the following problem: $$ \begin{cases} -\Delta u = \lambda a(x) f(u), & \text{in } \mathbb{R}^N, \\ u(x) \to 0, & \text{as } |x| \to +\infty. \end{cases} $$where $N≥3$, $λ$ is a real parameter and $a∈C_{loc}^α$ ($\mathbb{R}^N$,$\mathbb{R}$) for some $α∈(0,1)$ is a weighted function, $f: \mathbb{R}$→$\mathbb{R}$ is a Hölder continuous function with exponent $α$ such that $f(s)s$>0 for any $s\ne0$. We determine the intervals of $λ$ for the existence, exact multiplicity of one-sign solutions for this problem. We use bifurcation techniques and the approximation of connected components to prove our main results.
subject: Articles
  topic: section
subject: Unilateral global bifurcation
  topic: keywords
subject: one-sign solutions
  topic: keywords
subject: semilinear elliptic problems
  topic: keywords
subject: exact multiplicity
  topic: keywords
creator: Wenguo Shen
rights: Copyright (c) 2025 Journal of Partial Differential Equations https://creativecommons.org/licenses/by-nc-nd/4.0
date: 2025-12-27
publication date: 2025-12-27
starting page: 476
ending page: 493
doi: 10.4208/jpde.v38.n4.6

Several Parameter Type Logarithmic Sobolev Inequalities
type: item
description: In this paper, we obtain a parameter type logarithmic Sobolev inequality with weight and a modified parameter type logarithmic Sobolev inequality with weight on Euclidean space based on the parameter type logarithmic Caffarelli-Kohn-Nirenberg inequality on Euclidean space, respectively. By virtue of the convexity of a special function equivalent to a logarithmic Hölder inequality, and combining the Sobolev in-equality and Gagliardo-Nirenberg inequality on Hörmander’s vector fields, we also derive a logarithmic Sobolev inequality and a parameter type logarithmic Gagliardo-Nirenberg inequality on Hörmander’s vector fields, respectively. In addition, a parameter type logarithmic Sobolev inequality on Hörmander’s vector fields is given by suitable stretching transformation.
subject: Articles
  topic: section
subject: Logarithmic Sobolev inequality
  topic: keywords
subject: parameter type
  topic: keywords
subject: weight
  topic: keywords
subject: Euclidean space
  topic: keywords
subject: Hörmander’s vector fields
  topic: keywords
creator: Tingfu Feng
creator: Yan Dong
creator: Yan Zhu
creator: Gui Mu
rights: Copyright (c) 2025 Journal of Partial Differential Equations https://creativecommons.org/licenses/by-nc-nd/4.0
date: 2025-12-27
publication date: 2025-12-27
starting page: 462
ending page: 475
doi: 10.4208/jpde.v38.n4.5

Multiplicity of Solutions for a Class of Critical Choquard Equation in a Bounded Domain
type: item
description: In this paper, we consider the following critical Choquard equation$$-\Delta u = \mu f(x)|u|^{p-2}u + \left( \int_{\Omega} \frac{g(y)|u(y)|^{6-v}}{|x-y|^v} dy \right) g(x)|u|^{4-v}u, \quad x \in \Omega,$$where $µ>0$ is a parameter, $ν∈(0,3), p∈(4,6)$ and $f ,g$ are continuous functions. For $µ$ small enough, by using Lusternik-Schnirelmann category theory, we establish a relationship between the number of solutions and the category of the global maximum set of $g$.
subject: Articles
  topic: section
subject: Choquard equation
  topic: keywords
subject: category theory
  topic: keywords
subject: critical exponent
  topic: keywords
subject: multiple solutions
  topic: keywords
creator: Yongpeng Chen
creator: Baoxia Jin
rights: Copyright (c) 2025 Journal of Partial Differential Equations https://creativecommons.org/licenses/by-nc-nd/4.0
date: 2025-12-27
publication date: 2025-12-27
starting page: 446
ending page: 461
doi: 10.4208/jpde.v38.n4.4

Existence of Multiple Weak Solutions for Nonlinear Problems on the Sierpiński Gasket
type: item
description: In this paper, we study the existence of at least two, three and infinitely many solutions for nonlinear problems on the Sierpiński gasket, modelling some physical phenomena such as reaction-diffusion problems, elastic properties of fractal media and flow through fractal non-smooth domains. We will obtain the existence of two weak solutions when nonlinear term $f(x,t)$ is non-negative, when it is non-positive in neighbored of zero and otherwise is positive we will show the existence of three weak solutions, and when it is odd we will get the existence of infinitely many solutions. The results are proved by using some critical point theorems.
subject: Articles
  topic: section
subject: Sierpiński gasket
  topic: keywords
subject: fractal domains
  topic: keywords
subject: nonlinear elliptic equation
  topic: keywords
subject: variational methods
  topic: keywords
subject: critical point
  topic: keywords
creator: Ahmad Ghobadi
creator: Shapour Heidarkhani
rights: Copyright (c) 2025 Journal of Partial Differential Equations https://creativecommons.org/licenses/by-nc-nd/4.0
date: 2025-12-27
publication date: 2025-12-27
starting page: 432
ending page: 445
doi: 10.4208/jpde.v38.n4.3

Multiple Periodic Solutions of Allen-Cahn System Involving Fractional Laplacian
type: item
description: We consider periodic solutions of the following nonlinear system associated with the fractional Laplacian $$(−∂_{xx})^su(x)+∇F(u(x))=0 \quad {\rm in} \quad \mathbb{R},$$ where $F$ : $\mathbb{R}^2$→$\mathbb{R}$ is a smooth double-well potential. For the case that $F$ is even in its two variables we obtain the existence of more and more periodic solutions with large period, by using Clark’s theorem. For the case that $F$ is only even in its the second variable and the origin is a saddle critical point of $F$, we give two periodic solutions by using Morse theory.
subject: Articles
  topic: section
subject: Multiple periodic solutions
  topic: keywords
subject: fractional Laplacian
  topic: keywords
subject: Allen-Cahn system
  topic: keywords
subject: Clark’s theorem
  topic: keywords
subject: Morse theory
  topic: keywords
creator: Zhuoran Du
creator: Zhenping Feng
rights: Copyright (c) 2025 Journal of Partial Differential Equations https://creativecommons.org/licenses/by-nc-nd/4.0
date: 2025-12-27
publication date: 2025-12-27
starting page: 411
ending page: 431
doi: 10.4208/jpde.v38.n4.2

Uniform Attractors for Non-Autonomous Reaction-Diffusion Equation with Mixed Delay
type: item
description: This paper is concerned with the asymptotic behavior of solutions of non-autonomous reaction-diffusion equation with delays. The well-posedness theory of equation for the initial data belonging to $C_{{L^r}(Ω)}$($1< r < ∞$) and $C_{{W^{1,r}}(Ω)}$($1< r < N$) is established respectively. In addition, the existence of uniform attractors in $C_{{L^r}(Ω)}$ for the family of processes with translation bounded external force is proved. Moreover, the long time behavior of solution with higher regularity in $C_{{W^{1,r}}(Ω)}$ is considered as well.
subject: Articles
  topic: section
subject: Uniform attractors
  topic: keywords
subject: reaction-diffusion equation with delays
  topic: keywords
subject: non-autonomous equation
  topic: keywords
creator: Yanping Ran
creator: Xiaoke Sun
creator: Yueli Liu
creator: Maolin Liang
creator: Zhiqiang Hu
rights: Copyright (c) 2025 Journal of Partial Differential Equations https://creativecommons.org/licenses/by-nc-nd/4.0
date: 2025-12-27
publication date: 2025-12-27
starting page: 377
ending page: 410
doi: 10.4208/jpde.v38.n4.1

A Food Chain Model with a Protection Zone
type: item
description: In this paper, we are concerned about a food chain model with a protection zone for the prey species. Dynamical behavior, nonexistence and existence of positive steady states are obtained and there exist several critical values determined by the parameters and the protection zone for the growth rate of the prey species. The results reveal that the protection zone is effective for the survival of the prey species and beneficial for the coexistence of multiple species. Moreover, different properties of positive steady states from those of the two-species models are shown. The introduction of the prey or the top predator can be either favorable or unfavorable for the coexistence of multiple species.
subject: Articles
  topic: section
subject: Dynamical behavior
  topic: keywords
subject: positive steady state
  topic: keywords
subject: food chain model
  topic: keywords
subject: protection zone.
  topic: keywords
subject: 35J55
  topic: subjects
subject: 35B32
  topic: subjects
subject: 92D25
  topic: subjects
creator: Yuxia Wang
creator: Yifu Dong
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2025-09-15
publication date: 2025-09-15
starting page: 352
ending page: 375
doi: 10.4208/jpde.v38.n3.7

Fine Behaviors of Eigenvalues and Eigenfunctions of the Lane-Emden Problem in Dimension Two
type: item
description: Recently, qualitative analysis of peaked solutions of Lane-Emden problem in dimension two has been widely considered. In particular, the Morse index of concentrated solutions with a single peak or multi peaks has been computed in [15,16] separately. In this paper, we continue to consider the qualitative properties of the eigenvalues and eigenfunctions of the linearized Lane-Emden problem associated to peak solutions. Here we establish the fine behaviors of the first $m$ eigenvalues and eigenfunctions of the linearized Lane-Emden problem in dimension two, and correspondingly the number of concentrated points of the first $m$ eigenfunctions are given.
subject: Articles
  topic: section
subject: Lane-Emden problem
  topic: keywords
subject: eigenfunctions and eigenvalues
  topic: keywords
subject: asymptotic behavior.
  topic: keywords
subject: 35A02
  topic: subjects
subject: 35J08
  topic: subjects
subject: 35J60
  topic: subjects
creator: Kefan Pan
creator: Shuying Tian
creator: Pingping Yang
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2025-09-15
publication date: 2025-09-15
starting page: 335
ending page: 351
doi: 10.4208/jpde.v38.n3.6

Global Large Solution for the Tropical Climate Model Without Thermal Diffusion
type: item
description: In this paper, we consider the Cauchy problem of $d$-dimensional $(d=2,3)$ tropical climate model without thermal diffusion and construct global smooth solutions by choosing a class of special initial data $(u_0,v_0,θ_0)$ whose $L^2$ norm can be arbitrarily large, and improve the previous results in [1,2].
subject: Articles
  topic: section
subject: Tropical climate model
  topic: keywords
subject: zero thermal diffusion
  topic: keywords
subject: global large solution
  topic: keywords
subject: Sobolev spaces.
  topic: keywords
subject: 35D35
  topic: subjects
subject: 76D03
  topic: subjects
creator: Mulan Zhou
creator: Yanghai Yu
creator: Xiaolei Yang
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2025-09-15
publication date: 2025-09-15
starting page: 309
ending page: 323
doi: 10.4208/jpde.v38.n3.4

Global and Non-Global Solutions for Pseudo-Parabolic Equation with Singular Potential
type: item
description: This paper considers initial boundary value to a pseudo-parabolic equation with singular potential $\frac{u_t}{|x|^s}-\Delta u_t-\Delta u =|u|^{p-2}u$ with $2<p<\frac{2N}{N−2},$ which was studied in [1] by Lian et al. They dealt with the global existence, asymptotic behavior with low initial level $J(u_0)≤d$ and got the blow-up conditions of solutions with low and high initial level. In this paper, we give a new blow-up result which independent of the initial Nehari functional $I(u_0)$, and estimate the lower bound for blow-up time under some conditions. Finally, the precise exponential decay estimate is obtained for global solution with some conditions.
subject: Articles
  topic: section
subject: Pseudo-parabolic equation
  topic: keywords
subject: singular potential
  topic: keywords
subject: blow-up
  topic: keywords
subject: bounds for blow up time
  topic: keywords
subject: exponential decay.
  topic: keywords
subject: 35K70
  topic: subjects
subject: 35B44
  topic: subjects
creator: Chunxiao Yang
creator: Xinyu Pan
creator: Yuqing Chen
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press https://creativecommons.org/licenses/by-nc-nd/4.0
date: 2025-09-15
publication date: 2025-09-15
starting page: 324
ending page: 334
doi: 10.4208/jpde.v38.n3.5

Boundary Regularity for an Even Order Elliptic System in the Critical Dimension
type: item
description: In this short note, we consider the Dirichlet problem associated to an even order elliptic system with antisymmetric first order potential. Given any continuous boundary data, we show that weak solutions are continuous up to boundary.
subject: Articles
  topic: section
subject: Polyharmonic maps
  topic: keywords
subject: higher order elliptic system
  topic: keywords
subject: boudary continuity
  topic: keywords
subject: Dirichlet problem.
  topic: keywords
subject: 35J48
  topic: subjects
subject: 35B65
  topic: subjects
subject: 35G35
  topic: subjects
creator: Minglun Liu
creator: Yaolan Tian
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2025-09-15
publication date: 2025-09-15
starting page: 302
ending page: 308
doi: 10.4208/jpde.v38.n3.3

Asymptotic Behavior in a Quasilinear Two-Species Chemotaxis System with Nonlinear Sensitivity and Nonlinear Signal Secretion
type: item
description: In this paper, we study the asymptotic behavior of solutions to a quasilinear two-species chemotaxis system with nonlinear sensitivity and nonlinear signal production under homogeneous Neumann boundary conditions in a smooth bounded domain $Ω⊂\mathbb{R}^n(n≥2)$ where the parameter $\mu_1,\mu_2,γ_1,γ_2$ are positive constants, $τ ∈ \{0,1\}.$ The diffusion coefficients $D_i,Si ∈ C^2([0,∞))$ satisfy $D_i(s)≥a_0(s+1)^{−m_i},$ $0≤S_i(s)≤ b_0s(s+1)^{β_i−1},$ $s≥0,$ $m_i,β_i ∈\mathbb{R},$ $a_0,b_0 >0,$ $i=1,2.$ Under the assumption of properly initial data regularity, we can find appropriate $\mu_i$ such that the globally bounded solution of this system satisfies the following relationship. (I) If $a_1,a_2∈(0,1)$ and $\mu_1$ and $\mu_2$ are sufficiently large, then any globally bounded solution exponentially converges to $(\frac{1-a_1}{1-a_1a_2}, (\frac{1-a_2}{1-a_1a_2})^{\gamma_1}, \frac{1-a_2}{1-a_1a_2}, (\frac{1-a_1}{1-a_1a_2})^{\gamma_2})$ as $t→∞;$ (II) If $a_1 >1>a_2 >0$ and $\mu_2$ is sufficiently large, then any globally bounded solution exponentially converges to (0,1,1,0) as $t→∞;$ (III) If $a_1=1>a_2 >0$ and $\mu_2$ is sufficiently large, then any globally bounded solution algebraically converges to (0,1,1,0) as $t→∞.$
subject: Articles
  topic: section
subject: Two-species chemotaxis
  topic: keywords
subject: asymptotic behavior
  topic: keywords
subject: exponential convergence
  topic: keywords
subject: algebraical convergence.
  topic: keywords
subject: 35K55
  topic: subjects
subject: 32O92
  topic: subjects
subject: 35B35
  topic: subjects
subject: 92C17
  topic: subjects
creator: Nannan Zhao
creator: Zhongping Li
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2025-09-15
publication date: 2025-09-15
starting page: 279
ending page: 301
doi: 10.4208/jpde.v38.n3.2

Estimates on Weighted $L^q$-Norms of the Nonstationary 3D Navier-Stokes Flow in an Exterior Domain
type: item
description: This paper is devoted to estimates on weighted $L^q$-norms of the nonstationary 3D Navier-Stokes flow in an exterior domain. By multiplying the Navier-Stokes equation with a well selected vector field, an integral equation is derived, from which, we establish the weighted estimate $∥|x|^αu(t)∥_q ≤ C(1+t^{\frac{α}{2} +ε} )t^{-\frac{3}{2}(1-\frac{1}{q})} ,$$t>0,$ where $ 0<α≤1$ and $\frac{3}{2}0$ holds, where $α>0$ and $1
subject: Articles
  topic: section
subject: weighted estimates
  topic: keywords
subject: navier-stokes flow
  topic: keywords
subject: exterior domain
  topic: keywords
subject: $L^1$-data.
  topic: keywords
subject: 35Q30
  topic: subjects
subject: 76D05
  topic: subjects
creator: Qinghua Zhang
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2025-09-15
publication date: 2025-09-15
starting page: 251
ending page: 278
doi: 10.4208/jpde.v38.n3.1

Parabolic System Related to the P-Laplician with Degeneracy on the Boundary
type: item
description: " In this article, we study the system with boundary degeneracy<\/p> $u_{it}-{\\rm div}(a(x)|\\triangledown u_{i}|^{p_{i}-2}\\nabla u_i)=f_{i}(x,t,u_1,u_2),\\qquad (x,t)\\in\\Omega_T$.<\/p> Applying the monotone iterattion technique and the regularization method, we get the existence of solution for a regularized system. Moreover, under an integral condition on the coefficient function $a(x)$, % And if %$ \\int_{\\Omega} a(x)^{-\\frac{1}{min{(p_1,p_2)}-1}} {\\rm d}x{\\rm d}t\\leq C ,$ the existence and the uniqueness of the local solutions of the system is obtained by using a standard limiting process. Finally, the stability of the solutions is proved without any boundary value condition, provided $a(x)$ satisfies another restriction.<\/p>"
subject: Articles
  topic: section
subject: Weak solution;boundary degeneracy parabolic system;initial boundary value problem;existence
  topic: keywords
subject: stability.
  topic: keywords
subject: 35B40
  topic: subjects
subject: 35K65
  topic: subjects
subject: 35K55
  topic: subjects
creator: Qitong Ou
creator: Huashui Zhan
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2019-10-14
publication date: 2019-10-14
starting page: 281
ending page: 292
doi: 10.4208/jpde.v32.n3.5

Structural Stability of p(x)-Laplace Problems with Fourier Type Boundary Condition
type: item
description: " We study the continuous dependence on coefficients of solutions of the nonlinear nonhomogeneous Fourier boundary value problems involving the p(x)-Laplace\r\noperator.<\/p>"
subject: Articles
  topic: section
subject: Generalized Lebesgue and Sobolev spaces;Leray-Lions operator;weak solution;renormalized solution;Thermorheological fluids;continuous dependence;Fourier type boundary condition;Young measures.
  topic: keywords
subject: 35J60
  topic: subjects
subject: 35D05
  topic: subjects
subject: 76A05
  topic: subjects
creator: Kpe Kansie
creator: Stanislas\u0003 Ouaro
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2019-10-14
publication date: 2019-10-14
starting page: 229
ending page: 268
doi: 10.4208/jpde.v32.n3.3

The Liouville Type Theorem for a System of Nonlinear Integral Equations on Exterior Domain
type: item
description: " In this paper we are concerned with a system of nonlinear integral equations\r\non the exterior domain under the suitable boundary conditions. Through the method\r\nof moving planes in integral forms which has some innovative ideas we obtain that\r\nthe exterior domain is radial symmetry and a pair of positive solutions of the system\r\nis radial symmetry and monotone non-decreasing. Consequently, we can obtain the\r\ncorresponding Liouville type theorem about the solutions.<\/p>"
subject: Articles
  topic: section
subject: System of integral equations;exterior domain;symmetry;monotonicity;Liouville type theorem.
  topic: keywords
subject: 35A15
  topic: subjects
creator: Rong Yin
creator: Jihui Zhang
creator: Xudong Shang
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2019-10-14
publication date: 2019-10-14
starting page: 191
ending page: 206
doi: 10.4208/jpde.v32.n3.1

A Second-order Hyperbolic Chemotaxis Model
type: item
description: " In this paper, we study a hyperbolic type chemotaxis model in one space dimension. We assume the speed is constant, the production and degradation of the external signal $s$ is given by \\mbox{$-\\beta s+f(u^++u^-)$,} where $f(u^++u^-)$ is the general form and $u^+, u^-$ depend on $s$. The existence of the weak solution of the model considered in the paper is obtained by the method of characteristics and the contraction mapping principle.<\/p>"
subject: Articles
  topic: section
subject: Chemotaxis model;hyperbolic equations;correlated random walks.
  topic: keywords
subject: 35L45
  topic: subjects
subject: 35M10
  topic: subjects
subject: 92C17
  topic: subjects
creator: Shaohua Wu
creator: Haiying Chen
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2019-10-14
publication date: 2019-10-14
starting page: 269
ending page: 280
doi: 10.4208/jpde.v32.n3.4

Bifurcation and Stability Analysis in Complex Cross-Diffusion Mathematical Model of Phytoplankton-Fish Dynamics
type: item
description: " In this paper, we propose a nonlinear reaction-diffusion system describing\r\nthe interaction between toxin-producing phytoplankton and fish population. We analyze the effect of cross-diffusion on the dynamics of the system. The mathematical\r\nstudy of the model leads us to have an idea on the existence of a solution, the existence\r\nof equilibria and the stability of the stationary equilibria. Finally, numerical simulations performed at two-dimensions allowed us to establish the formation of spatial\r\npatterns and a threshold of release of the toxin, above which we talk about the\r\nphytoplankton blooms.<\/p>"
subject: Articles
  topic: section
subject: Toxin effect;populations dynamics;predator-prey model;reaction-diffusion system;bifurcation;pattern formation.
  topic: keywords
subject: 65P30
  topic: subjects
subject: 68Q85
  topic: subjects
subject: 68U20
  topic: subjects
creator: Hamidou Ouedraogo
creator: Wendkouni Ouedraogo
creator: Boureima Sangar\u00e9
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2019-10-14
publication date: 2019-10-14
starting page: 207
ending page: 228
doi: 10.4208/jpde.v32.n3.2

Transportation Inequalities for Multivalued Stochastic Evolution Equations
type: item
description: " In this paper, using the Girsanov transformation argument, we establish Talagrand-type T_2 inequalities under the d_2 metric and the uniform metric d_∞ for the law of the solution of a class multivalued stochastic evolution equations."
subject: Articles
  topic: section
subject: Transportation cost inequalities;Girsanov transformation;multivalued stochastic evolution equation
  topic: keywords
subject: 60H15
  topic: subjects
subject: 60G15
  topic: subjects
subject: 60H05
  topic: subjects
creator: Liping Xu
creator: Zhi Li
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2017-08-02
publication date: 2017-08-02
starting page: 254
ending page: 263
doi: 10.4208/jpde.v30.n3.5

Mixed Layer Problem of a Three-Dimensional Drift-Diffusion Model for Semiconductors
type: item
description: " The quasineutral limit and the mixed layer problem of a three-dimensional drift-diffusion model is discussed in this paper. For the Neumann boundaries and the general initial data, the quasineutral limit is proven rigorously with the help of the weighted energy method, the matched asymptotic expansion method of singular perturbation problem and the entropy production inequality."
subject: Articles
  topic: section
subject: Drift-diffusion model;quasineutral limit;mixed layer
  topic: keywords
subject: 35B25
  topic: subjects
subject: 35B40
  topic: subjects
subject: 35K57
  topic: subjects
creator: Chundi Liu
creator: Shu Wang
creator: Wenqing Xu
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2017-08-02
publication date: 2017-08-02
starting page: 264
ending page: 280
doi: 10.4208/jpde.v30.n3.6

Solutions to a 3D Burgers Equation with Initial Discontinuity That Are Two Disjoint Spheres
type: item
description: " We study the singular structure of a kind of three dimensional non-selfsimilar global solutions and their interaction for quasilinear hyperbolic conservation laws. The initial discontinuity is two disjoint unit spheres and initial data just contain two different constant states, the global solutions and some new phenomena are discovered. We give the solutions in 0 < t < T^* and T^* < t, and at t=T^*, the two basic shock waves and the constant state u_ are disappeared.Then, we find a new shock wave between two rarefaction by R-H condition. Finally, we show the limit of the solution when t → ∞. A technique is proposed to construct the three dimensional shock wave without dimensional reduction or coordinate transformation."
subject: Articles
  topic: section
subject: Initial discontinuity;Burgers equation;elementary wave;global solution
  topic: keywords
subject: 35L65
  topic: subjects
subject: 35L67
  topic: subjects
creator: Shu Wang
creator: Niu Haiping
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2017-08-02
publication date: 2017-08-02
starting page: 232
ending page: 253
doi: 10.4208/jpde.v30.n3.4

Entropy Solution to Nonlinear Elliptic Problem with Non-local Boundary Conditions and L¹-data
type: item
description: " We study a nonlinear elliptic problem with non-local boundary conditions and L¹-data. We prove an existence and uniqueness result of an entropy solution.<\/p>"
subject: Articles
  topic: section
subject: Entropy solution;non-local boundary conditions;Leray-Lions operator
  topic: keywords
subject: 35J05
  topic: subjects
subject: 35J25
  topic: subjects
subject: 35J60
  topic: subjects
subject: 35J66
  topic: subjects
creator: Stanislas\u0003 Ouaro
creator: Safimba Soma
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2017-08-02
publication date: 2017-08-02
starting page: 201
ending page: 220
doi: 10.4208/jpde.v30.n3.2

Semi-linear Elliptic Equations on Graph
type: item
description: " Let G=(V,E) be a locally finite graph, Ω ⊂ V be a finite connected set, Δ be the graph Laplacian, and suppose that h : V → R is a function satisfying the coercive condition on Ω, namely there exists some constant δ > 0 such that $$∫_Ωu(-Δ+h)udμ ≥ δ ∫_Ω|∇u|²dμ,\\quad \u2200u:V → R.$$ By the mountain-pass theoremof Ambrosette-Rabinowitz, we prove that for any p > 2, there exists a positive solution to $$-Δu+hu=|u|^{p-2}u\\quad\\;\\; in\\;\\; Ω$$. Using the same method, we prove similar results for the p-Laplacian equations. This partly improves recent results of Grigor'yan-Lin-Yang."
subject: Articles
  topic: section
subject: Sobolev embedding;Yamabe type equation;Laplacian on graph
  topic: keywords
subject: 34B45
  topic: subjects
subject: 35A15
  topic: subjects
subject: 58E30
  topic: subjects
creator: Dongshuang Zhang
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2017-08-02
publication date: 2017-08-02
starting page: 221
ending page: 231
doi: 10.4208/jpde.v30.n3.3