Journal of Partial Differential Equations
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.
Free Transport Equation and Hyperbolic Schrödinger Equation via Wigner Transformation
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.
Copyright (c) 2026 Journal of Partial Differential Equations https://creativecommons.org/licenses/by-nc-nd/4.0
Regularity Criteria for 3D Liquid Crystal Flows in Besov Space
This note is devoted to investigating regularity criteria for 3D liquid crystal flows in Besov space.
Spectral Properties of the Sub-Laplacian on 2-Step Stratified Lie Groups Without the Moore-Wolf Condition
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.
On Concentration of Real Solutions for Fractional Helmholtz Equation
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 $Q(x).$
Monotonic Behavior of Positive Solutions for Semi-Linear Parabolic Equations with Uniformly Elliptic Non-Local Operators in Half-Space
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.
Eigenvalue Problem for a Class of Nonlinear Operators Containing $p$(·)-Laplacian in a Variable Exponent Sobolev Space
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.
Well-Posedness of 5th-Order KdV Equation Posed on a Finite Domain with Nonlinear Boundary Values
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.
Copyright (c) 2025 Journal of Partial Differential Equations https://creativecommons.org/licenses/by-nc-nd/4.0
The Existence and Exact Multiplicity of One-Sign Solutions for Semilinear Elliptic Problems in $\mathbb{R}^N$
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.
Several Parameter Type Logarithmic Sobolev Inequalities
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.
Multiplicity of Solutions for a Class of Critical Choquard Equation in a Bounded Domain
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 $g$.
Existence of Multiple Weak Solutions for Nonlinear Problems on the Sierpiński Gasket
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.
Multiple Periodic Solutions of Allen-Cahn System Involving Fractional Laplacian
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.
Uniform Attractors for Non-Autonomous Reaction-Diffusion Equation with Mixed Delay
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.
A Food Chain Model with a Protection Zone
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.
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Fine Behaviors of Eigenvalues and Eigenfunctions of the Lane-Emden Problem in Dimension Two
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.
Global Large Solution for the Tropical Climate Model Without Thermal Diffusion
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].
Global and Non-Global Solutions for Pseudo-Parabolic Equation with Singular Potential
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.
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Boundary Regularity for an Even Order Elliptic System in the Critical Dimension
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.
Asymptotic Behavior in a Quasilinear Two-Species Chemotaxis System with Nonlinear Sensitivity and Nonlinear Signal Secretion
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 $t→∞.$
Estimates on Weighted $L^q$-Norms of the Nonstationary 3D Navier-Stokes Flow in an Exterior Domain
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
Parabolic System Related to the P-Laplician with Degeneracy on the Boundary
" 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)$, %\u00a0 And if\u00a0 %$ \\int_{\\Omega}\u00a0 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>"
Weak solution;boundary degeneracy parabolic system;initial boundary value problem;existence
Structural Stability of p(x)-Laplace Problems with Fourier Type Boundary Condition
" 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>"
Generalized Lebesgue and Sobolev spaces;Leray-Lions operator;weak solution;renormalized solution;Thermorheological fluids;continuous dependence;Fourier type boundary condition;Young measures.
The Liouville Type Theorem for a System of Nonlinear Integral Equations on Exterior Domain
" 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>"
System of integral equations;exterior domain;symmetry;monotonicity;Liouville type theorem.
A Second-order Hyperbolic Chemotaxis Model
" 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\u00a0 the method of characteristics and the contraction mapping principle.<\/p>"
Chemotaxis model;hyperbolic equations;correlated random walks.
Bifurcation and Stability Analysis in Complex Cross-Diffusion Mathematical Model of Phytoplankton-Fish Dynamics
" 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>"
Toxin effect;populations dynamics;predator-prey model;reaction-diffusion system;bifurcation;pattern formation.
Transportation Inequalities for Multivalued Stochastic Evolution Equations
" 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_\u221e for the law of the solution of a class multivalued stochastic evolution equations."
Transportation cost inequalities;Girsanov transformation;multivalued stochastic evolution equation
Mixed Layer Problem of a Three-Dimensional Drift-Diffusion Model for Semiconductors
" 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."
Drift-diffusion model;quasineutral limit;mixed layer
Solutions to a 3D Burgers Equation with Initial Discontinuity That Are Two Disjoint Spheres
" 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 \u2039 t \u2039 T^* and T^* \u2039 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 \u2192 \u221e. A technique is proposed to construct the three dimensional shock wave without dimensional reduction or coordinate transformation."
Initial discontinuity;Burgers equation;elementary wave;global solution
Entropy Solution to Nonlinear Elliptic Problem with Non-local Boundary Conditions and L¹-data
" We study a nonlinear elliptic problem with non-local boundary conditions and L\u00b9-data. We prove an existence and uniqueness result of an entropy solution.<\/p>"
Entropy solution;non-local boundary conditions;Leray-Lions operator
" Let G=(V,E) be a locally finite graph, \u03a9 \u2282 V be a finite connected set, \u0394 be the graph Laplacian, and suppose that h : V \u2192 R is a function satisfying the coercive condition on \u03a9, namely there exists some constant \u03b4 \u203a 0 such that $$\u222b_\u03a9u(-\u0394+h)ud\u03bc \u2265 \u03b4 \u222b_\u03a9|\u2207u|\u00b2d\u03bc,\\quad \u2200u:V \u2192 R.$$ By the mountain-pass theoremof Ambrosette-Rabinowitz, we prove that for any p \u203a 2, there exists a positive solution to $$-\u0394u+hu=|u|^{p-2}u\\quad\\;\\; in\\;\\; \u03a9$$. Using the same method, we prove similar results for the p-Laplacian equations. This partly improves recent results of Grigor'yan-Lin-Yang."
Sobolev embedding;Yamabe type equation;Laplacian on graph