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: 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: Entropy Solution to Nonlinear Elliptic Problem with Non-local Boundary Conditions and L¹-data
items: Solutions to a 3D Burgers Equation with Initial Discontinuity That Are Two Disjoint Spheres
items: Mixed Layer Problem of a Three-Dimensional Drift-Diffusion Model for Semiconductors
items: Semi-linear Elliptic Equations on Graph
items: Transportation Inequalities for Multivalued Stochastic Evolution Equations
items: Constructing Infinite Sequence Solutions for Space-Time Fractional Benjamin-Bona-Mahoney Equation
items: Asymptotic Behavior of the Solution to a 3-D Simplified Energy-Transport Model for Semiconductors
items: Infinite Sequence Solutions for Space-Time Fractional Symmetric Regularized Long Wave Equation
items: Non-local Problem for the Loaded Integral-differential Equation in Double-connected Domain
items: Inequalities and Separation for a Biharmonic Laplace-Beltrami Differential Operator in a Hilbert Space Associated with the Existence and Uniqueness Theorem
items: Regularity of Viscous Solutions for a Degenerate Non-linear Cauchy Problem
items: Asymptotic Behavior for a Viscoelastic Wave Equation with a Time-varying Delay Term
items: On a Linear Partial Differential Equation of the Higher Order in Two Variables with Initial Condition Whose Coefficients are Real-valued Simple Step Functions
items: An Alternative Method for Solving Lagrange's First-Order Partial Differential Equation with Linear Function Coefficients
items: Remarks on Liouville Type Result for the 3D Hall-MHD System
items: Conservation Laws for CKdV and BSSK Systems
items: Renormalized Solutions of Nonlinear Parabolic Equations in Weigthed Variable-Exponent Space
items: Random Attractor for the Nonclassical Diffusion Equation with Fading Memory

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 $a(x)$, %\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>"
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\u00a0 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

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 L\u00b9-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

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 \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."
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

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

Semi-linear Elliptic Equations on Graph
type: item
description: " 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."
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

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_\u221e 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

Constructing Infinite Sequence Solutions for Space-Time Fractional Benjamin-Bona-Mahoney Equation
type: item
description: " In this article, the space-time fractional Benjamin-Bona-Mahoney equation is investigated via applying the first kind of elliptic equation method. As a result, based on the B\u00e4cklund transformations and some seed solutions of the first kind of elliptic equation, families of infinite sequence solutions are presented. This approach is also applicable to many other nonlinear fractional differential equations."
subject: Articles
  topic: section
subject: Space-time fractional Benjamin-Bona-Mahoney equation;Bäcklund transformations;infinite sequence solutions
  topic: keywords
subject: 35A24
  topic: subjects
subject: 35R11
  topic: subjects
subject: 35-04
  topic: subjects
creator: Zhouzheng Kang
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2017-08-02
publication date: 2017-08-02
starting page: 189
ending page: 200
doi: 10.4208/jpde.v30.n3.1

Asymptotic Behavior of the Solution to a 3-D Simplified Energy-Transport Model for Semiconductors
type: item
description: " The well-posedness of smooth solution to a 3-Dsimplified Energy-Transport model is discussed in this paper. We prove the local existence, uniqueness, and asymptotic behavior of solution to the equations with hybrid cross-diffusion. The smooth solution convergences to a stationary solution with an exponential rate as time tends to infinity when the initial date is a small perturbation of the stationary solution."
subject: Articles
  topic: section
subject: Energy-Transport model;Gagliardo-Nirenberg inequality;asymptotic behavior
  topic: keywords
subject: 42C40
  topic: subjects
subject: 42C15
  topic: subjects
creator: Chundi Liu
creator: Yong Li
creator: Shu Wang
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2016-03-05
publication date: 2016-03-05
starting page: 71
ending page: 88
doi: 10.4208/jpde.v29.n1.7

Infinite Sequence Solutions for Space-Time Fractional Symmetric Regularized Long Wave Equation
type: item
description: " In this paper, we investigate the space-time fractional symmetric regularized long wave equation. By using the B\u00e4cklund transformations and nonlinear superposition formulas of solutions to Riccati equation, we present infinite sequence solutions for space-time fractional symmetric regularized long wave equation. This method can be extended to solve other nonlinear fractional partial differential equations."
subject: Articles
  topic: section
subject: Space-time fractional symmetric regularized long wave equation;Bäcklund transformations;nonlinear superposition formulas;exact solutions
  topic: keywords
subject: 35A24
  topic: subjects
subject: 35R11
  topic: subjects
subject: 35-04
  topic: subjects
creator: Zhouzheng Kang
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2016-03-05
publication date: 2016-03-05
starting page: 48
ending page: 58
doi: 10.4208/jpde.v29.n1.5

Non-local Problem for the Loaded Integral-differential Equation in Double-connected Domain
type: item
description: " In this work an existence and uniqueness of solution of the non-local boundary value problemfor the loaded elliptic-hyperbolic type equation with integral-differential operations in double-connected domain have been investigated. The uniqueness of solution is proved by the method of integral energy using an extremum principle for the mixed type equations, and the existence is proved by the method of integral equations."
subject: Articles
  topic: section
subject:
  topic: keywords
  value: Loaded equation;elliptic-hyperbolic type;integral-differential operators;doubleconnected domain;an extremum principle;existence of solution;uniqueness of solution;method of integral energy;integral equations
subject: 35M10
  topic: subjects
creator: O. Kh Abdullayev
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2016-03-05
publication date: 2016-03-05
starting page: 36
ending page: 47
doi: 10.4208/jpde.v29.n1.4

Inequalities and Separation for a Biharmonic Laplace-Beltrami Differential Operator in a Hilbert Space Associated with the Existence and Uniqueness Theorem
type: item
description: " In this paper, we have studied the separation for the biharmonic Laplace-Beltrami differential operator<\/p> \\begin{equation*}Au(x)=-\\Delta \\Delta u(x)+V(x)u(x),\\end{equation*}<\/p> for all $x\\in R^{n}$, in the Hilbert space $H=L_{2}(R^{n},H_{1})$ with the operator potential $V(x)\\in C^{1}(R^{n},L(H_{1}))$, where $L(H_{1})$ is the space of all bounded linear operators on the Hilbert space $H_{1}$, while $\\Delta \\Delta u$\\ is the biharmonic differential operator and<\/p> \\begin{equation*}\\Delta u{=-}\\sum_{i,j=1}^{n}\\frac{1}{\\sqrt{\\det g}}\\frac{\\partial }{{\\partial x_{i}}}\\left[ \\sqrt{\\det g}g^{-1}(x)\\frac{\\partial u}{{\\partial x}_{j}}\\right]\\end{equation*}<\/p> is the Laplace-Beltrami differential operator in $R^{n}$. Here $g(x)=(g_{ij}(x))$ is the Riemannian matrix, while $g^{-1}(x)$ is the inverse of the matrix $g(x)$. Moreover, we have studied the existence and uniqueness Theorem for the solution of the non-homogeneous biharmonic Laplace-Beltrami differential equation $Au=-\\Delta \\Delta u+V(x)u(x)=f(x)$ in the Hilbert space $H$ where $f(x)\\in H$ as an application of the separation approach.<\/p> <br\/><\/p>"
subject: Articles
  topic: section
subject: Separation biharmonic Laplace-Beltrami operator;operator potential;Hilbert space L_2(R^n;H_1);coercive estimate;existence and uniqueness Theorem
  topic: keywords
subject: 47F05
  topic: subjects
subject: 58J99
  topic: subjects
creator: Elsayed M. E. Zayed
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2016-03-05
publication date: 2016-03-05
starting page: 59
ending page: 70
doi: 10.4208/jpde.v29.n1.6

Regularity of Viscous Solutions for a Degenerate Non-linear Cauchy Problem
type: item
description: " We consider the Cauchy problem for a class of nonlinear degenerate parabolic equation with forcing. By using the vanishing viscosity method it is possible to construct a generalized solution. Moreover, this solution is a Lipschitz function on the spatial variable and H\u00f6lder continuous with exponent 1\/2 on the temporal variable."
subject: Articles
  topic: section
subject: Viscosity solution;Hölder estimates;Hölder continuity
  topic: keywords
subject: 35K65
  topic: subjects
subject: 35D10
  topic: subjects
creator: Eric Hern\u00e1ndez Sastoque
creator: Christian Klingenberg
creator: Leonardo Rend\u00f3n
creator: Juan C. Juajibioy
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2016-03-05
publication date: 2016-03-05
starting page: 14
ending page: 21
doi: 10.4208/jpde.v29.n1.2

Asymptotic Behavior for a Viscoelastic Wave Equation with a Time-varying Delay Term
type: item
description: " The following viscoelastic wave equation with a time-varying delay term in internal feedback $|u_t|^\u03c1u_{tt}-\u0394u-\u0394u_{tt}+\u222b^t_0g(t-s)\u0394u(s)ds+\u03bc_1u_t(x,t)+\u03bc_2u_t(x,t-\u03c4(t))=0$, is considered in a bounded domain. Under appropriate conditions on \u03bc_1, \u03bc_2 and on the kernel g, we establish the general decay result for the energy by suitable Lyapunov functionals."
subject: Articles
  topic: section
subject: Global existence;asymptotic behavior;general decay;time-varying delay
  topic: keywords
subject: 35L05
  topic: subjects
subject: 35L15
  topic: subjects
subject: 35L70
  topic: subjects
subject: 93D15
  topic: subjects
creator: Shun Tang Wu
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2016-03-05
publication date: 2016-03-05
starting page: 22
ending page: 35
doi: 10.4208/jpde.v29.n1.3

On a Linear Partial Differential Equation of the Higher Order in Two Variables with Initial Condition Whose Coefficients are Real-valued Simple Step Functions
type: item
description: " By using the method developed in the paper [Georg. Inter. J. Sci. Tech., Volume 3, Issue 1 (2011), 107-129], it is obtained a representation in an explicit form of the weak solution of a linear partial differential equation of the higher order in two variables with initial condition whose coefficients are real-valued simple step functions."
subject: Articles
  topic: section
subject: Linear partial differential equation of the higher order in two variables;Fourier differential operator
  topic: keywords
subject: 34Axx
  topic: subjects
subject: 34A35
  topic: subjects
subject: 34K06
  topic: subjects
creator: Gogi Pantsulaia
creator: Givi Giorgadze
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2016-03-05
publication date: 2016-03-05
starting page: 1
ending page: 13
doi: 10.4208/jpde.v29.n1.1

An Alternative Method for Solving Lagrange's First-Order Partial Differential Equation with Linear Function Coefficients
type: item
description: " An alternative method of solving Lagrange's first-order partial differential equation of the form $$(a_1x+b_1y+c_1z)p+(a_2x+b_2y+c_2z)q=a_3x+b_3y+c_3z,$$ where p=\u2202z\/\u2202x, q=\u2202z\/\u2202y and a_i, b_i, c_i (i=1,2,3) are all real numbers has been presented here."
subject: Articles
  topic: section
subject: Lagrange's first-order partial differential equation;linear functions;simultaneous ordinary differential equations;linear homogeneous algebraic equations
  topic: keywords
subject: 35A25
  topic: subjects
subject: 35A99
  topic: subjects
creator: Syed Md Himayetul Islam
creator: J. Das
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2015-09-05
publication date: 2015-09-05
starting page: 208
ending page: 224
doi: 10.4208/jpde.v28.n3.2

Remarks on Liouville Type Result for the 3D Hall-MHD System
type: item
description: " In this paper, we consider the 3D Hall-MHD system, and provide an improved Liouville type result for its stationary version."
subject: Articles
  topic: section
subject: Hall-MHD system;Liouville theorem
  topic: keywords
subject: 35Q35
  topic: subjects
subject: 35B65
  topic: subjects
creator: Zujin Zhang
creator: Xian Yang
creator: Shulin Qiu
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2015-09-05
publication date: 2015-09-05
starting page: 286
ending page: 290
doi: 10.4208/jpde.v28.n3.7

Conservation Laws for CKdV and BSSK Systems
type: item
description: " In current paper, the coupled KdV (CKdV) system and Bosonized Supersymmetric Sawada-Kotera (BSSK) system are considered. Some linearly independent conservation laws for the two systems are derived via the first homotopy approach and symbolic computation."
subject: Articles
  topic: section
subject: Coupled KdV system;Bosonized Supersymmetric Sawada-Kotera system;conservation laws
  topic: keywords
subject: 35A25
  topic: subjects
subject: 35G20
  topic: subjects
creator: Zhouzheng Kang
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2015-09-05
publication date: 2015-09-05
starting page: 278
ending page: 285
doi: 10.4208/jpde.v28.n3.6

Renormalized Solutions of Nonlinear Parabolic Equations in Weigthed Variable-Exponent Space
type: item
description: " This article is devoted to study the existence of renormalized solutions for the nonlinear p(x)-parabolic problemin the Weighted-Variable-Exponent Sobolev spaces, without the sign condition and the coercivity condition."
subject: Articles
  topic: section
subject: Weighted variable exponent Lebesgue Sobolev space;Young's inequality;renormalized solution;parabolic problems
  topic: keywords
subject: 35J15
  topic: subjects
subject: 35J70
  topic: subjects
subject: 35J85
  topic: subjects
creator: Youssef Akdim
creator: Chakir Allalou
creator: El gorch Nezha
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2015-09-05
publication date: 2015-09-05
starting page: 225
ending page: 252
doi: 10.4208/jpde.v28.n3.3

Random Attractor for the Nonclassical Diffusion Equation with Fading Memory
type: item
description: " In this paper,we consider the stochastic nonclassical diffusion equationwith fading memory on a bounded domain. By decomposition of the solution operator, we give the necessary condition of asymptotic smoothness of the solution to the initial boundary value problem, and then we prove the existence of a random attractor in the space $M_1=D(A^{\\frac{1}{2}}) \u00d7 L^2_\u03bc(R^+, D(A^{\\frac{1}{2}}))$, where A=-\u0394 with Dirichlet boundary condition."
subject: Articles
  topic: section
subject: Stochastic nonclassical diffusion equations;fading memory;random attractor
  topic: keywords
subject: 35B40
  topic: subjects
subject: 35B41
  topic: subjects
subject: 35K55
  topic: subjects
creator: Shuilin Cheng
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2015-09-05
publication date: 2015-09-05
starting page: 253
ending page: 268
doi: 10.4208/jpde.v28.n3.4