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On the well-posedness in Besov–Herz spaces for the inhomogeneous incompressible Euler equations Dyn. Partial Differ. Equ. (IF 1.3) Pub Date : 2023-11-07 Lucas C. F. Ferreira, Daniel F. Machado
In this paper we study the inhomogeneous incompressible Euler equations in the whole space $\mathbb{R}^n$ with $n \geq 3$. We obtain well-posedness and blow-up results in a new framework for inhomogeneous fluids, more precisely Besov–Herz spaces that are Besov spaces based on Herz ones, covering particularly critical cases of the regularity. Comparing with previous works on Besov spaces, our results
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Liouville theorems for nonnegative solutions to weighted Schrödinger equations with logarithmic nonlinearities Dyn. Partial Differ. Equ. (IF 1.3) Pub Date : 2023-11-07 Yuxia Guo, Shaolong Peng
In this paper, we are concerned with the physically interesting static weighted Schrödinger equations involving logarithmic nonlinearities:\[(-\Delta)^s u = c_1 {\lvert x \rvert}^a u^{p_1} \log (1 + u^{q_1}) + c_2 {\lvert x \rvert}^b\Bigl( \dfrac{1}{{\lvert \: \cdot \: \rvert}^\sigma} \Bigr) u^{p_2} \quad \textrm{,}\]where $n \geq 2, 0 \lt s =: m + \frac{\alpha}{2} \lt +\infty , 0 \lt \alpha \leq 2
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On a parabolic-elliptic Keller–Segel system with nonlinear signal production and nonlocal growth term Dyn. Partial Differ. Equ. (IF 1.3) Pub Date : 2023-11-07 Pan Zheng
and nonlocal growth term\[\begin{cases}u_t = \Delta u - \chi \nabla \cdot (u^m \nabla v) + u \Biggl( a_0 - a_1 u^\alpha + a_2 \displaystyle \int_\Omega u^\sigma dx \Biggr) & (x, t) \in \Omega \times (0,\infty) \; , \\0=\Delta - v + u^\gamma , & (x, t) \in \Omega \times (0,\infty) \; , \\\end{cases}\]under homogeneous Neumann boundary conditions in a smoothly bounded domain $\Omega \subset \mathbb{R}^n
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Global well-posedness to the 3D Cauchy problem of nonhomogeneous micropolar fluids involving density-dependent viscosity with large initial velocity and micro-rotational velocity Dyn. Partial Differ. Equ. (IF 1.3) Pub Date : 2023-11-07 Ling Zhou, Chun-Lei Tang
We show the global well-posedness to the three-dimensional (3D) Cauchy problem of nonhomogeneous micropolar fluids with density-dependent viscosity and vacuum in $\mathbb{R}^3$ provided that the initial mass is sufficiently small. Moreover, we also obtain that the gradients of velocity and micro-rotational velocity converge exponentially to zero in $H^1$ as time goes to infinity. Our analysis relies
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Threshold solutions for the 3D focusing cubic-quintic nonlinear Schrödinger equation at low frequencies Dyn. Partial Differ. Equ. (IF 1.3) Pub Date : 2023-11-01 Masaru Hamano, Hiroaki Kikuchi, Minami Watanabe
This paper addresses the focusing cubic-quintic nonlinear Schrödinger equation in three space dimensions. Especially, we study the global dynamics of solutions whose energy and mass equal to those of the ground state in the spirits of Duyckaerts and Merle $\href{ https://doi.org/10.1007/s00039-009-0707-x}{[14]}$. When we try to obtain the corresponding results of [14], we meet several difficulties
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Energy conservation and Onsager’s conjecture for a surface growth model Dyn. Partial Differ. Equ. (IF 1.3) Pub Date : 2023-11-01 Wei Wei, Yulin Ye, Xue Mei
In this paper, it is shown that the energy equality of weak solution $v$ to a surface growth model is valid if $v_x \in L^p (0, T; L^q(\mathbb{T}))$ with $\frac{3}{p} + \frac{1}{q} = 1$ and $1 \leq q \leq 4$, or $v \in L^\infty (0, T; L^\infty (\mathbb{T}))$, or $v_{xx} \in L^p (0, T; L^q (\mathbb{T}))$ with $\frac{2}{p} + \frac{2}{5q} = 1$ and $q \geq 1$, which gives an affirmative answer to a question
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Periodic and quasi-periodic Euler-$\alpha$ flows close to Rankine vortices Dyn. Partial Differ. Equ. (IF 1.3) Pub Date : 2023-11-01 Emeric Roulley
In the present contribution, we first prove the existence of $\mathbf{m}$-fold simply-connected V‑states close to the unit disc for Euler-$\alpha$ equations. These solutions are implicitly obtained as bifurcation curves from the circular patches. We also prove the existence of quasi-periodic in time vortex patches close to the Rankine vortices provided that the scale parameter $\alpha$ belongs to a
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Global regularity of multi-dimensional Burgers equation with critical dissipation only in one direction Dyn. Partial Differ. Equ. (IF 1.3) Pub Date : 2023-11-01 Zhuan Ye
In this paper, we are concerned with the multi-dimensional Burgers equation with the critical dissipation only in one direction. We make use of the elegant method introduced by Constantin and Vicol to show the unique global existence of smooth solution.
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Unique continuation results for abstract quasi-linear evolution equations in Banach spaces Dyn. Partial Differ. Equ. (IF 1.3) Pub Date : 2023-05-19 Igor Leite Freire
Unique continuation properties for a class of evolution equations defined on Banach spaces are considered from two different point of views: the first one is based on the existence of conserved quantities, which very often translates into the conservation of some norm of the solutions in a suitable Banach space. The second one considers well-posed problems. Our results are then applied to some equations
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Local well-posedness and regularity criterion for nonhomogeneous magneto-micropolar fluid equations without angular viscosity Dyn. Partial Differ. Equ. (IF 1.3) Pub Date : 2023-05-19 Jishan Fan, Xin Zhong
We study an initial-boundary-value problem for three-dimensional nonhomogeneous magneto-micropolar fluid equations without angular viscosity. Using linearization and Banach’s fixed point theorem, we prove the local existence and uniqueness of strong solutions. Moreover, a regularity criterion is also obtained.
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Blow-up or Grow-up for the threshold solutions to the nonlinear Schrödinger equation Dyn. Partial Differ. Equ. (IF 1.3) Pub Date : 2023-05-19 Stephen Gustafson, Takahisa Inui
We consider the nonlinear Schrödinger equation with $L^2$-supercritical and $H^1$-subcritical power type nonlinearity. Duyckaerts and Roudenko [8] and Campos, Farah, and Roudenko [3] studied the global dynamics of the solutions with same mass and energy as that of the ground state. In these papers, finite variance is assumed to show the finite time blow-up. In the present paper, we remove the finite-variance
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Convergence to steady states of parabolic sine-Gordon Dyn. Partial Differ. Equ. (IF 1.3) Pub Date : 2023-05-19 Min Gao, Jiao Xu
Based on the recent surprising work on the symmetry breaking phenomenon of the Allen–Cahn equation [11, 12], we consider the one-dimensional parabolic sine‑Gordon equation with periodic boundary conditions. Particularly, we derive a strong dependence of the non-trivial steady states on the diffusion coefficient $\kappa$ and provide some description on them for $0 \lt \kappa \lt 1$. To further investigate
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Analyticity of the semigroup corresponding to a strongly damped wave equation with a Ventcel boundary condition Dyn. Partial Differ. Equ. (IF 1.3) Pub Date : 2023-05-19 Mehdi Badra, Takéo Takahashi
We consider a wave equation with a structural damping coupled with an undamped wave equation located at its boundary. We prove that, due to the coupling, the full system is parabolic. In order to show that the underlying operator generates an analytical semigroup, we study in particular the effect of the damping of the “interior” wave equation on the “boundary” wave equation and show that it generates
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Explicit solutions of atmospheric Ekman flows for some eddy viscosities in ellipsoidal coordinates Dyn. Partial Differ. Equ. (IF 1.3) Pub Date : 2023-05-17 Taoyu Yang, Michal Fečkan, Jinrong Wang
In ellipsoidal coordinates, we study the motion of the wind in the steady atmospheric Ekman layer for the height-dependent eddy viscosities in the form of some quadratic, fourth and rational power functions. We construct the explicit solutions for these forms of the eddy viscosities by using suitable boundary conditions. Furthermore, we write down a formula of the angle between the wind vector and
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Global strong solution and exponential decay to the 3D incompressible Bénard system with density-dependent viscosity and vacuum Dyn. Partial Differ. Equ. (IF 1.3) Pub Date : 2023-05-17 Min Liu, Yong Li
In this paper, we study the Cauchy problem of the incompressible Bénard system with density-dependent viscosity on the whole three-dimensional space. We first construct a key priori exponential estimates by the energy method, and then we prove that there is a unique global strong solution for the 3D Cauchy problem under the assumption that initial energy is suitably small. In particular, it is not
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Polymer kinetic theory temperature dependent configurational probability diffusion equations: Existence of positive solution results Dyn. Partial Differ. Equ. (IF 1.3) Pub Date : 2023-05-17 Ionel Sorin Ciuperca, Liviu Iulian Palade
A new configurational probability diffusion—CPD—equation that accounts for temperature dependent molecular dynamics for incompressible polymer fluids is introduced. We prove the existence of positive solutions for the corresponding variational formulation using Schauder’s fixed point theory. The polymer fluid macromolecules are modeled as Finitely Extensible Nonlinear Elastic dumbbells, also known
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Singular Levy processes and dispersive effects of generalized Schrödinger equations Dyn. Partial Differ. Equ. (IF 1.3) Pub Date : 2023-05-17 Yannick Sire, Xueying Yu, Haitian Yue, Zehua Zhao
We introduce new models for Schrödinger-type equations, which generalize standard NLS and for which different dispersion occurs depending on the directions. Our purpose is to understand dispersive properties depending on the directions of propagation, in the spirit of waveguide manifolds, but where the diffusion is of different types. We mainly consider the standard Euclidean space and the waveguide
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Analytic regularity for Navier–Stokes–Korteweg model on pseudo-measure spaces Dyn. Partial Differ. Equ. (IF 1.3) Pub Date : 2022-12-23 A. Tendani Soler
The purpose of this work is to study the existence and analytic smoothing effect for the compressible Navier–Stokes system with quantum pressure in pseudo-measure spaces. This system has been considered by B. Haspot and an analytic smoothing effect for a Korteweg-type system was considered by F. Charve, R. Danchin and J. Xu, both of them in Besov spaces. Here we give a better lower bound of the radius
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Flowbox theorems for a class of Sobolev vector fields Dyn. Partial Differ. Equ. (IF 1.3) Pub Date : 2022-12-23 Mário Bessa
We give sufficient conditions for the flowbox theorem for Sobolev vector fields to be valid not only for the general case but also for the conservative case.
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Dynamics of subcritical threshold solutions for energy-critical NLS Dyn. Partial Differ. Equ. (IF 1.3) Pub Date : 2022-12-23 Qingtang Su, Zehua Zhao
In this paper, we study the dynamics of subcritical threshold solutions for focusing energy critical NLS on $\mathbb{R}^d \, (d \geq 5)$ with nonradial data. This problem with radial assumption was studied by T. Duyckaerts and F. Merle in [19] for $d = 3, 4, 5$ and later by D. Li and X. Zhang in [25] for $d \geq 6$. We generalize the conclusion for the subcritical threshold solutions by removing the
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Applications of Nijenhuis geometry IV: Multicomponent KdV and Camassa–Holm equations Dyn. Partial Differ. Equ. (IF 1.3) Pub Date : 2022-12-23 Alexey V. Bolsinov, Andrey Yu. Konyaev, Vladimir S. Matveev
We construct a new series of multi-component integrable PDE systems that contains as particular examples (with appropriately chosen parameters) and generalises many famous integrable systems including KdV, coupled KdV [1], Harry Dym, coupled Harry Dym [2], Camassa–Holm, multicomponent Camassa–Holm [14], Dullin–Gottwald–Holm and Kaup–Boussinesq systems. The series also contains integrable systems with
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Dynamic analysis of a diffusive eco-epidemiological system with fear effect and prey refuge Dyn. Partial Differ. Equ. (IF 1.3) Pub Date : 2022-12-14 Tingting Ma, Xinzhu Meng
In the evolutionary development of species, the prey can produce the fear effect in the face of predation behaviors. This fear effect may affect the own reproduction growth of the prey. In order to reduce the risk of predation, the prey has the instinct to protect themselves. At the same time, the population is easy vulnerable by disease in the ecosystem. Driven by these biological facts, we propose
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Asymptotic behavior of global solutions to some multidimensional quasilinear hyperbolic systems Dyn. Partial Differ. Equ. (IF 1.3) Pub Date : 2022-12-14 Dongbing Zha, Minghui Sun
For the Cauchy problem of multidimensional quasilinear hyperbolic systems of diagonal form without self-interaction, the global existence of classical solutions with small initial data was shown in [13]. In this paper, we will first prove that the global solution will scatter to free linear waves in some weighted $L^p$ sense, then based on it, we will study the rigidity aspect of scattering problem
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Global existence and asymptotic behavior of solutions for a fractional chemotaxis-Navier-Stokes system Dyn. Partial Differ. Equ. (IF 1.3) Pub Date : 2022-12-14 Miguel A. Fontecha-Medina, Élder J. Villamizar-Roa
We consider a fractional chemotaxis-Navier-Stokes model in the whole space $\mathbb{R}^N , N \geq 2$, with a time-fractional variation in the Caputo sense, a fractional self-diffusion for the physical variables and a fractional dissipation mechanism for the chemoattraction process. We prove the existence and uniqueness of global mild solutions with small initial data in a larger class of critical spaces
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Constant vorticity atmospheric Ekman flows in the modified $\beta$-plane approximation Dyn. Partial Differ. Equ. (IF 1.3) Pub Date : 2022-12-14 Yi Guan, Michal Fečkan, Jinrong Wang
In this paper, we study the classical problem of the wind in the steady atmospheric Ekman layer with constant eddy viscosity. The full nonlinear governing equations with the general boundary conditions are considered in the sense of modified $\beta$‑plane approximation. Under the assumption of a flat surface and constant vorticity vector, we show that the flow has one nonvanishing component, a result
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Global regularity of magneto-micropolar equations with logarithmically dissipation Dyn. Partial Differ. Equ. (IF 1.3) Pub Date : 2022-12-14 Jingbo Wu, Xiaoqin Nan, Yan Jia
Inspired by a work of Terence Tao on a generalized Navier–Stokes equations (T. Tao, Anal. PDE 2009), we examine in this paper the global regularity of the 3D magneto-micropolar equations with logarithmically hyperdissipative velocity dissipation.
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Dual Lyapunov approach to finite time stability for parabolic PDE Dyn. Partial Differ. Equ. (IF 1.3) Pub Date : 2022-05-23 Anna Michalak, Andrzej Nowakowski
We investigate stability and finite time stability properties of the zero solution to a semilinear parabolic equation. To this effect we develop a new, dual approach to Lyapunov concept of stability. The dual Lyapunov function satisfies a dual Hamilton–Jacobi inequality. This is a basis to study finite time stability (finite extinction time) of the original problem.
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Inviscid limit of the inhomogeneous incompressible Navier–Stokes equations under the weak Kolmogorov hypothesis in $\mathbb{R}^3$ Dyn. Partial Differ. Equ. (IF 1.3) Pub Date : 2022-05-23 Dixi Wang, Cheng Yu, Xinhua Zhao
In this paper, we consider the inviscid limit of inhomogeneous incompressible Navier–Stokes equations under the weak Kolmogorov hypothesis in $\mathbb{R}^3$. In particular, this limit is a weak solution of the corresponding Euler equations. We first deduce the Kolmogorov-type hypothesis in $\mathbb{R}^3$, which yields the uniform bounds of $\alpha^\mathit{th}$‑order fractional derivatives of $\sqrt{\rho^\mu}
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Asymptotic behavior of short trajectories to nonhomogeneous heat-conducting magnetohydrodynamic equations Dyn. Partial Differ. Equ. (IF 1.3) Pub Date : 2022-05-23 Pigong Han, Keke Lei, Chenggang Liu, Xuewen Wang
In this paper, we study the asymptotic behavior of short trajectories of weak solutions to the 2D nonhomogeneous heat-conducting magnetohydrodynamic equations. Several bounds for short trajectories are obtained. An attracting set is constructed, which consists of orbits on [0, 1] of complete bounded solutions. Furthermore, the attracting set is compact in different topologies.
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Asymptotics toward rarefaction wave for an inflow problem of the compressible Navier–Stokes–Korteweg equation Dyn. Partial Differ. Equ. (IF 1.3) Pub Date : 2022-05-23 Yeping Li, Yujie Qian, Shengqi Yu
In this article, we are concerned with the large-time behavior of solutions to an inflow problem in one-dimensional case for the Navier–Stokes–Korteweg equation, which models compressible fluids with internal capillarity. We first investigate that the asymptotic state is the rarefaction wave under the proper condition of the far fields and boundary values. The asymptotic stability of the rarefaction
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Traveling waves of a generalized nonlinear Beam equation Dyn. Partial Differ. Equ. (IF 1.3) Pub Date : 2022-05-19 Amin Esfahani, Steven Levandosky
We consider the existence and stability of traveling waves of a nonlinear beam equation for a general class of non-homogeneous nonlinearities. We use variational methods to prove existence of ground state traveling wave solutions for this class and analyze their stability. We also present a numerical method based on the variational characterization of ground states and use it to determine intervals
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Global wellposedness for 2D quasilinear wave without Lorentz Dyn. Partial Differ. Equ. (IF 1.3) Pub Date : 2022-05-19 Xinyu Cheng, Dong Li, Jiao Xu, Dongbing Zha
We consider the two-dimensional quasilinear wave equations with standard null-form type quadratic nonlinearities. We introduce a new streamlined framework and prove global wellposedness without using the Lorentz boost vector fields.
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An elliptic nonlinear system of multiple functions with application Dyn. Partial Differ. Equ. (IF 1.3) Pub Date : 2022-05-19 Joon Hyuk Kang, Timothy Robertson
The purpose of this paper is to give a sufficient conditions for the existence and uniqueness of positive solutions to a rather general type of elliptic system of the Dirichlet problem on a bounded domain $\Omega$ in $R^n$. Also considered are the effects of perturbations on the coexistence state and uniqueness. The techniques used in this paper are super‑sub solutions method, eigenvalues of operators
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A remark on the Strichartz inequality in one dimension Dyn. Partial Differ. Equ. (IF 1.3) Pub Date : 2022-05-19 Ryan Frier, Shuanglin Shao
In this paper, we study the extremal problem for the Strichartz inequality for the Schrödinger equation on $\mathbb{R}^2$. We show that the solutions to the associated Euler–Lagrange equation are exponentially decaying in the Fourier space and thus can be extended to be complex analytic. Consequently we provide a new proof to the characterization of the extremal functions: the only extremals are Gaussian
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On blow-up solutions to the nonlinear Schrödinger equation in the exterior of a convex obstacle Dyn. Partial Differ. Equ. (IF 1.3) Pub Date : 2021-12-02 Oussama Landoulsi
In this paper, we consider the Schrödinger equation with a masssupercritical focusing nonlinearity, in the exterior of a smooth, compact, convex obstacle of $\mathbb{R}^d$ with Dirichlet boundary conditions. We prove that solutions with negative energy blow up in finite time. Assuming furthermore that the nonlinearity is energy-subcritical, we also prove (under additional symmetry conditions) blow-up
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On the well-posedness of the incompressible Euler equations in a larger space of Besov–Morrey type Dyn. Partial Differ. Equ. (IF 1.3) Pub Date : 2021-12-02 Lucas C. F. Ferreira, Jhean E. Pérez-López
We obtain a local-in-time well-posedness result and blow-up criterion for the incompressible Euler equations in a new framework, namely Besov spaces based on modified weak-Morrey spaces, covering critical and supercritical cases of the regularity. In comparison with some previous results and considering the same level of regularity, we provide a larger initial-data class for the well-posedness of the
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Global strong solution to the Cauchy problem of 1D viscous two-fluid model without any domination condition Dyn. Partial Differ. Equ. (IF 1.3) Pub Date : 2021-12-02 Xiaona Gao, Zhenhua Guo, Zilai Li
In this paper, we consider the Cauchy problem to the compressible two-fluid Navier–Stokes equations in one-dimensional space allowing vacuum. It is shown that the compressible two-fluid Navier–Stokes equations admit global strong solution with the large initial value and no the domination condition1 which was posed in [A. Vasseur, H.Y. Wen, C. Yu, J. Math. Pure. Appl. 125 (2019), 247–282], when the
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Generalized Strichartz estimates for wave and Dirac equations in Aharonov–Bohm magnetic fields Dyn. Partial Differ. Equ. (IF 1.3) Pub Date : 2021-12-02 Federico Cacciafesta, Zhiqing Yin, Junyong Zhang
We prove generalized Strichartz estimates for wave and massless Dirac equations in Aharonov–Bohm magnetic fields. Following a well established strategy to deal with scaling critical perturbations of dispersive PDEs, we make use of Hankel transform and rely on some precise estimates on Bessel functions. As a complementary result, we prove a local smoothing estimate for the Klein–Gordon equation in the
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An invasive-invaded species dynamics with a high order diffusion operator Dyn. Partial Differ. Equ. (IF 1.3) Pub Date : 2021-12-02 José Palencia
The introduction of the Landau–Ginzburg free energy provides a framework to generalize the diffusion beyond the classical fickian approach. The analysis shows the existence and uniqueness of solutions with a priori bounds and making use of the Fixed Point Theorem to a suitable abstract evolution. Asymptotic solutions are provided with the Hamilton–Jacobi operator and a positivity condition is formulated
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Observable measures in partial differential equations Dyn. Partial Differ. Equ. (IF 1.3) Pub Date : 2021-12-02 M. Dong, W. Jung, C. A. Morales
We prove that a broad class of PDE’s including the reaction-diffusion and 2D Navier–Stokes equations have observable measures. Moreover, these measures form the minimal weak* compact subset of Borel probability measures whose basins of attraction has total Gaussian measure. To prove these results we extend the theory of observable measures [8] from continuous maps on compact manifolds to dissipative
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The simplified Bardina equation on two-dimensional closed manifolds Dyn. Partial Differ. Equ. (IF 1.3) Pub Date : 2021-12-02 Pham Truong Xuan
In this paper we study the viscous simplified Bardina equation on the two-dimensional closed manifold $M$ which is embedded in $\mathbb{R}^3$. First, we prove the existence and the uniqueness of the weak solutions and also the existence of the global attractor for the equation on $M$. Then we establish the upper and lower bounds of the Hausdorff and fractal dimensions of the global attractor. We also
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The Liouville type theorem for the stationary magnetohydrodynamic equations in weighted mixed-norm Lebesgue spaces Dyn. Partial Differ. Equ. (IF 1.3) Pub Date : 2021-12-02 Huiying Fan, Meng Wang
In this paper, we are concentrated on demonstrating the Liouville type theorem for the stationary Magnetohydrodynamic equations in mixednorm Lebesgue spaces and weighted mixed-norm Lebesgue spaces. In particular, we show that, under some sufficient conditions in (weighted) mixed-norm Lebesgue spaces, the solution of stationary MHDs are identically zero. Precisely, we investigate solutions of MHDs that
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$W^{1,\infty}$ instability of $H^1$-stable peakons in the Novikov equation Dyn. Partial Differ. Equ. (IF 1.3) Pub Date : 2021-07-22 Robin Ming Chen, Dmitry E. Pelinovsky
Peakons in the Novikov equation have been proved to be orbitally and asymptotically stable in $H^1$. Meanwhile, it is also known that the $H^1$ topology is ill-suited for the local well-posedness theory. In this paper we investigate the stability property under the stronger $W^{1,\infty}$ topology where these peakons belong to and the local well-posedness theory can be established. We prove that the
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A new approach to study constant vorticity water flows in the $\beta$-plane approximation with centripetal forces Dyn. Partial Differ. Equ. (IF 1.3) Pub Date : 2021-07-22 Fahe Miao, Michal Fečkan, Jinrong Wang
In this paper, we study three-dimensional equatorial flows with constant vorticity beneath a wave train and above a flat bed in the $\beta$-plane approximation with centripetal forces by adopting higher order approximation about the Coriolis terms. We show that the new approximation about the Coriolis forces is also applicable for certain problem, which help us to derive the same result.
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Lions-type theorem of the fractional Laplacian and applications Dyn. Partial Differ. Equ. (IF 1.3) Pub Date : 2021-07-22 Zhaosheng Feng, Yu Su
In this paper, our goal is to establish a generalized version of Lions-type theorem for the fractional Laplacian. As an application of this theorem, we consider the existence of ground state solutions of a fractional equation:\[(-\Delta)^s u + V (\lvert x \rvert) u = f(u), \; x \in \mathbb{R}^N ,\]where $N \geqslant 3, s \in (\frac{1}{2}, 1), V$ is a singular potential with $\alpha \in (0, 2s) \cup
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Almost continuity of a pullback random attractor for the stochastic $g$-Navier–Stokes equation Dyn. Partial Differ. Equ. (IF 1.3) Pub Date : 2021-07-22 Yangrong Li, Shuang Yang
A pullback random attractor for a cocycle is a family of compact invariant attracting random sets $A(t, \theta_s \cdot)$, where $(t, s)$ is a point of the Euclid plane and $\theta$ is a group of measure-preserving transformations on a probability space. Under three conditions including the union closedness of the universe, the time-sample compactness of the PRA and the joint continuity of the cocycle
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Asymptotic behavior of global solutions to one-dimension quasilinear wave equations Dyn. Partial Differ. Equ. (IF 1.3) Pub Date : 2021-06-01 Mengni Li
The asymptotic behavior of solutions is a significant subject in the theory of wave equations. In this paper we are concerned with the asymptotic behavior of the unique global solution to the Cauchy problem for one-dimension quasilinear wave equations with null conditions. By applying the small-data-global-existence result and exploiting the strength of weights, we not only provide sharper convergence
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Global well-posedness for the fifth-order Kadomtsev–Petviashvili II equation in anisotropic Gevrey spaces Dyn. Partial Differ. Equ. (IF 1.3) Pub Date : 2021-06-01 Aissa Boukarou, Daniel Oliveira da Silva, Kaddour Guerbati, Khaled Zennir
We show that the fifth-order Kadomtsev–Petviashvili II equation is globally well-posed in an anisotropic Gevrey space, which complements earlier results on the well-posedness of this equation in anisotropic Sobolev spaces.
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Existence and symmetry of solutions to 2-D Schrödinger–Newton equations Dyn. Partial Differ. Equ. (IF 1.3) Pub Date : 2021-06-01 Daomin Cao, Wei Dai, Yang Zhang
In this paper, we consider the following 2-D Schrödinger–Newton equations\[-\Delta u + a(x)u + \frac{\gamma}{2\pi} (\operatorname{log}(\lvert \cdot \rvert) \ast {\lvert u \rvert}^p) {\lvert u \rvert}^{p-2} u = b {\lvert u \rvert}^{q-2} u \quad \textrm{in} \; \mathbb{R}^2 \; \textrm{,}\]where $a \in C(\mathbb{R}^2)$ is a $\mathbb{Z}^2$-periodic function with $\operatorname{inf}_{\mathbb{R}^2} a \gt
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A remark on attractor bifurcation Dyn. Partial Differ. Equ. (IF 1.3) Pub Date : 2021-06-01 Chunqiu Li, Desheng Li, Jintao Wang
In this paper we present some local dynamic bifurcation results in terms of invariant sets of nonlinear evolution equations. We show that if the trivial solution is an isolated invariant set of the system at the critical value $\lambda = \lambda_0$, then either there exists a one-sided neighborhood $I^{-}$ of $\lambda_0$ such that for each $\lambda \in I^{-}$, the system bifurcates from the trivial
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Ergodicity effects on transport-diffusion equations with localized damping Dyn. Partial Differ. Equ. (IF 1.3) Pub Date : 2021-03-01 Kaïs Ammari, Taoufik Hmidi
The main objective of this paper is to study the time decay of transport-diffusion equation with inhomogeneous localized damping in the multi-dimensional torus. The drift is governed by an autonomous Lipschitz vector field and the diffusion by the standard heat equation with small viscosity parameter $\nu$. In the first part we deal with the inviscid case and show some results on the time decay of
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Asymptotic behavior of solutions for a class of two-coupled nonlinear fractional Schrödinger equations Dyn. Partial Differ. Equ. (IF 1.3) Pub Date : 2021-03-01 Brahim Alouini
In the current issue, we consider two coupled weakly dissipative fractional Schrödinger equations with cubic nonlinearities that reads\[\begin{cases}u_t - i(-\Delta)^{\frac{\alpha}{2}} u + i ({\lvert u \rvert}^2 + {\lvert v \rvert}^2) u + \gamma u = f \\v_t - i(-\Delta)^{\frac{\alpha}{2}} v + i ({\lvert u \rvert}^2 + {\lvert v \rvert}^2) v + \delta v_x + \gamma v = g\end{cases}\]We will prove that
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Global dynamics of partly diffusive Hindmarsh–Rose equations in neurodynamics Dyn. Partial Differ. Equ. (IF 1.3) Pub Date : 2021-03-01 Chi Phan, Yuncheng You, Jianzhong Su
Global dynamics of the partly diffusive Hindmarsh–Rose equations as a new mathematical model in neurodynamics is presented and studied in this paper. The existence of global attractor for the solution semiflow is proved through uniform estimates showing the higher-order dissipative property and the ultimate compactness by the new approach of Kolmogorov–Riesz theorem.
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Strong solutions to the Cauchy problem of two-dimensional nonhomogeneous micropolar fluid equations with nonnegative density Dyn. Partial Differ. Equ. (IF 1.3) Pub Date : 2021-03-01 Xin Zhong
We consider the Cauchy problem of nonhomogeneous micropolar fluid equations with zero density at infinity on the whole space $\mathbb{R}^2$. By weighted energy method, we show the local existence and uniqueness of strong solutions provided that the initial density decays not too slowly at infinity.
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On endpoint regularity criterion of the 3D Navier–Stokes equations Dyn. Partial Differ. Equ. (IF 1.3) Pub Date : 2021-03-01 Zhouyu Li, Daoguo Zhou
Let $(u,\pi)$ with $u = (u_1, u_2, u_3)$ be a suitable weak solution of the three-dimensional Navier–Stokes equations in $\mathbb{R}^3 \times (0, T)$. Denote by $\dot{\mathcal{B}}^{-1}_{\infty,\infty}$ the closure of $C^\infty_0$ in $\dot{B}^{-1}_{\infty,\infty}$. We prove that if $u \in L^\infty (0, T; \dot{B}^{-1}_{\infty,\infty}), u(x, T) \in \dot{\mathcal{B}}^{-1}_{\infty,\infty})$, and $u_3 \in
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Equivalent definitions of Caputo derivatives and applications to subdiffusion equations Dyn. Partial Differ. Equ. (IF 1.3) Pub Date : 2020-12-01 Mykola Krasnoschok, Vittorino Pata, Sergii V. Siryk, Nataliya Vasylyeva
An equivalent definition of the fractional Caputo derivative $D^\nu_t g$, for $\nu \in (0, 1)$, is found, within suitable assumptions on $g$. Some applications to the fractional calculus and to the theory of fractional partial differential equations are then discussed. In particular, this alternative definition is used to prove the maximum principle for the classical solutions to the linear subdiffusion
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Exact and explicit internal water waves at arbitrary latitude with underlying currents Dyn. Partial Differ. Equ. (IF 1.3) Pub Date : 2020-01-01 Yanjuan Yang,Xun Wang
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Symmetric and uniform analytic solutions in phase space for Navier–Stokes equations Dyn. Partial Differ. Equ. (IF 1.3) Pub Date : 2020-01-01 Qixiang Yang
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On the strong solutions for a stochastic 2D nonlocal Cahn–Hilliard–Navier–Stokes model Dyn. Partial Differ. Equ. (IF 1.3) Pub Date : 2020-01-01 G. Deugoué,A. Ndongmo Ngana,T. Tachim Medjo
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Almost sure existence of global weak solutions to the Boussinesq equations Dyn. Partial Differ. Equ. (IF 1.3) Pub Date : 2020-01-01 Weinan Wang,Haitian Yue