Electronic Research Archive

March 2020 , Volume 28 , Issue 1

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Traveling waves for a nonlocal dispersal SIR model equipped delay and generalized incidence
Yang Yang, Yun-Rui Yang and Xin-Jun Jiao
2020, 28(1): 1-13 doi: 10.3934/era.2020001 +[Abstract](377) +[HTML](159) +[PDF](369.78KB)

In this paper, the existence and non-existence of traveling wave solutions are established for a nonlocal dispersal SIR model equipped delay and generalized incidence. In addition, the existence and asymptotic behaviors of traveling waves under critical wave speed are also contained. Especially, the boundedness of traveling waves is obtained completely without imposing additional conditions on the nonlinear incidence.

Global existence and propagation speed for a Degasperis-Procesi equation with both dissipation and dispersion Special Issues
Guenbo Hwang and Byungsoo Moon
2020, 28(1): 15-25 doi: 10.3934/era.2020002 +[Abstract](303) +[HTML](161) +[PDF](327.55KB)

In this paper, we consider the dissipative Degasperis-Procesi equation with arbitrary dispersion coefficient and compactly supported initial data. We establish the simple condition on the initial data which lead to guarantee that the solution exists globally. We also investigate the propagation speed for the equation under the initial data is compactly supported.

Blow-up criterion for the 3D viscous polytropic fluids with degenerate viscosities Special Issues
Yue Cao
2020, 28(1): 27-46 doi: 10.3934/era.2020003 +[Abstract](264) +[HTML](130) +[PDF](346.98KB)

In this paper, the Cauchy problem of the \begin{document}$ 3 $\end{document}D compressible Navier-Stokes equations with degenerate viscosities and far field vacuum is considered. We prove that the \begin{document}$ L^\infty $\end{document} norm of the deformation tensor \begin{document}$ D(u) $\end{document} (\begin{document}$ u $\end{document}: the velocity of fluids) and the \begin{document}$ L^6 $\end{document} norm of \begin{document}$ \nabla \log \rho $\end{document} (\begin{document}$ \rho $\end{document}: the mass density) control the possible blow-up of regular solutions. This conclusion means that if a solution with far field vacuum to the Cauchy problem of the compressible Navier-Stokes equations with degenerate viscosities is initially regular and loses its regularity at some later time, then the formation of singularity must be caused by losing the bound of \begin{document}$ D(u) $\end{document} or \begin{document}$ \nabla \log \rho $\end{document} as the critical time approaches; equivalently, if both \begin{document}$ D(u) $\end{document} and \begin{document}$ \nabla \log \rho $\end{document} remain bounded, a regular solution persists.

The existence of solutions for a shear thinning compressible non-Newtonian models Special Issues
Yukun Song, Yang Chen, Jun Yan and Shuai Chen
2020, 28(1): 47-66 doi: 10.3934/era.2020004 +[Abstract](255) +[HTML](139) +[PDF](413.61KB)

This paper is concerned with the initial boundary value problem for a shear thinning fluid-particle interaction non-Newtonian model with vacuum. The viscosity term of the fluid and the non-Newtonian gravitational force are fully nonlinear. Under Dirichlet boundary for velocity and the no-flux condition for density of particles, the existence and uniqueness of strong solutions is investigated in one dimensional bounded intervals.

Initial boundary value problem for a inhomogeneous pseudo-parabolic equation Special Issues
Jun Zhou
2020, 28(1): 67-90 doi: 10.3934/era.2020005 +[Abstract](306) +[HTML](152) +[PDF](777.08KB)

This paper deals with the global existence and blow-up of solutions to a inhomogeneous pseudo-parabolic equation with initial value \begin{document}$ u_0 $\end{document} in the Sobolev space \begin{document}$ H_0^1( \Omega) $\end{document}, where \begin{document}$ \Omega\subset \mathbb{R}^n $\end{document} (\begin{document}$ n\geq1 $\end{document} is an integer) is a bounded domain. By using the mountain-pass level \begin{document}$ d $\end{document} (see (14)), the energy functional \begin{document}$ J $\end{document} (see (12)) and Nehari function \begin{document}$ I $\end{document} (see (13)), we decompose the space \begin{document}$ H_0^1( \Omega) $\end{document} into five parts, and in each part, we show the solutions exist globally or blow up in finite time. Furthermore, we study the decay rates for the global solutions and lifespan (i.e., the upper bound of blow-up time) of the blow-up solutions. Moreover, we give a blow-up result which does not depend on \begin{document}$ d $\end{document}. By using this theorem, we prove the solution can blow up at arbitrary energy level, i.e. for any \begin{document}$ M\in \mathbb{R} $\end{document}, there exists \begin{document}$ u_0\in H_0^1( \Omega) $\end{document} satisfying \begin{document}$ J(u_0) = M $\end{document} such that the corresponding solution blows up in finite time.

Finite time blow-up for a wave equation with dynamic boundary condition at critical and high energy levels in control systems Special Issues
Xiaoqiang Dai, Chao Yang, Shaobin Huang, Tao Yu and Yuanran Zhu
2020, 28(1): 91-102 doi: 10.3934/era.2020006 +[Abstract](300) +[HTML](143) +[PDF](299.33KB)

We study the initial boundary value problem of linear homogeneous wave equation with dynamic boundary condition. We aim to prove the finite time blow-up of the solution at critical energy level or high energy level with the nonlinear damping term on boundary in control systems.

The Mahler measure of $ (x+1/x)(y+1/y)(z+1/z)+\sqrt{k} $
Huimin Zheng, Xuejun Guo and Hourong Qin
2020, 28(1): 103-125 doi: 10.3934/era.2020007 +[Abstract](250) +[HTML](128) +[PDF](417.89KB)

In this paper we study the Mahler measures of reciprocal polynomials \begin{document}$ (x+1/x)(y+1/y)(z+1/z)+\sqrt{k} $\end{document} for \begin{document}$ k = 16 $\end{document}, \begin{document}$ k = -104\pm60\sqrt{3} $\end{document}, \begin{document}$ 4096 $\end{document} and \begin{document}$ k = -2024\pm765\sqrt{7} $\end{document}. We prove six conjectural identities proposed by Samart in [16].

Decomposition of spectral flow and Bott-type iteration formula
Xijun Hu and Li Wu
2020, 28(1): 127-148 doi: 10.3934/era.2020008 +[Abstract](62) +[HTML](29) +[PDF](396.67KB)

Let \begin{document}$ A(t) $\end{document} be a continuous path of self-adjoint Fredholm operators, we derive a decomposition formula of spectral flow if the path is invariant under a matrix-like cogredient. As applications, we give the generalized Bott-type iteration formula for linear Hamiltonian systems.

On the existence of permutations conditioned by certain rational functions
Guo-Niu Han
2020, 28(1): 149-156 doi: 10.3934/era.2020009 +[Abstract](58) +[HTML](35) +[PDF](275.19KB)

We prove several conjectures made by Z.-W. Sun on the existence of permutations conditioned by certain rational functions. Furthermore, we fully characterize all integer values of the "inverse difference" rational function. Our proofs consist of both investigation of the mathematical properties of the rational functions and brute-force attack by computer for finding special permutations.

Ricci solitons of the $ \mathbb{H}^{2} \times \mathbb{R} $ Lie group
Lakehal Belarbi
2020, 28(1): 157-163 doi: 10.3934/era.2020010 +[Abstract](59) +[HTML](51) +[PDF](254.67KB)

In this work we consider the three-dimensional Lie group denoted by \begin{document}$ \mathbb{H}^{2} \times \mathbb{R} $\end{document}, equipped with left-invariant Riemannian metric. The existence of non-trivial (i.e., not Einstein) Ricci solitons on three-dimensional Lie group \begin{document}$ \mathbb{H}^{2} \times \mathbb{R} $\end{document} is proved. Moreover, we show that there are not gradient Ricci solitons.

Non-existence of solutions to some degenerate coercivity elliptic equations involving measures data
Maoji Ri, Shuibo Huang and Canyun Huang
2020, 28(1): 165-182 doi: 10.3934/era.2020011 +[Abstract](77) +[HTML](29) +[PDF](392.76KB)

In this paper, we main consider the non-existence of solutions \begin{document}$ u $\end{document} by approximation to the following quasilinear elliptic problem with principal part having degenerate coercivity:


where \begin{document}$ \Omega $\end{document} is a bounded smooth subset of \begin{document}$ \mathbb{R}^N(N>2) $\end{document}, \begin{document}$ 1<p<N $\end{document}, \begin{document}$ q>1 $\end{document}, \begin{document}$ 0\leq\theta<1 $\end{document}, \begin{document}$ \lambda $\end{document} is a measure which is concentrated on a set with zero \begin{document}$ r $\end{document} capacity \begin{document}$ (p<r\leq N) $\end{document}.

The anisotropic integrability logarithmic regularity criterion for the 3D MHD equations
Ahmad Mohammad Alghamdi, Sadek Gala, Chenyin Qian and Maria Alessandra Ragusa
2020, 28(1): 183-193 doi: 10.3934/era.2020012 +[Abstract](78) +[HTML](27) +[PDF](341.22KB)

This study is devoted to investigating the regularity criterion of the 3D MHD equations in terms of pressure in the framework of anisotropic Lebesgue spaces. The result shows that if a weak solution \begin{document}$ (u, b) $\end{document} satisfies


then \begin{document}$ (u, b) $\end{document} is regular at \begin{document}$ t = T $\end{document}, which improve the previous results on the MHD equations

A note on sign-changing solutions for the Schrödinger Poisson system
Hui Guo and Tao Wang
2020, 28(1): 195-203 doi: 10.3934/era.2020013 +[Abstract](82) +[HTML](33) +[PDF](332.0KB)

We consider the following nonlinear Schrödinger-Poisson system

where \begin{document}$ \lambda>0 $\end{document} and \begin{document}$ f $\end{document} is continuous. By combining delicate analysis and the method of invariant subsets of descending flow, we prove the existence and asymptotic behavior of infinitely many radial sign-changing solutions for odd \begin{document}$ f $\end{document}. The nonlinearity covers the case of pure power-type nonlinearity \begin{document}$ f(u) = |u|^{p-2}u $\end{document} with the less studied situation \begin{document}$ p\in(3, 4). $\end{document} This result extends and complements the ones in [Z. Liu, Z. Q. Wang, and J. Zhang, Ann. Mat. Pura Appl., 2016] from the coercive potential case to the constant potential case.

Global existence and energy decay of solutions for a wave equation with non-constant delay and nonlinear weights
Vanessa Barros, Carlos Nonato and Carlos Raposo
2020, 28(1): 205-220 doi: 10.3934/era.2020014 +[Abstract](82) +[HTML](28) +[PDF](347.49KB)

We consider the wave equation with a weak internal damping with non-constant delay and nonlinear weights given by

in a bounded domain. Under proper conditions on nonlinear weights $ \mu_1(t), \mu_2(t) $ and non-constant delay $ \tau(t) $, we prove global existence and estimative the decay rate for the energy.

Existence and uniform decay estimates for the fourth order wave equation with nonlinear boundary damping and interior source Special Issues
Huafei Di, Yadong Shang and Jiali Yu
2020, 28(1): 221-261 doi: 10.3934/era.2020015 +[Abstract](84) +[HTML](30) +[PDF](498.76KB)

In this paper, we consider the initial boundary value problem for the fourth order wave equation with nonlinear boundary velocity feedbacks \begin{document}$ f_{1}(u_{\nu{t}}) $\end{document}, \begin{document}$ f_{2}(u_{t}) $\end{document} and internal source \begin{document}$ |u|^{\rho}u $\end{document}. Under some geometrical conditions, the existence and uniform decay rates of the solutions are proved even if the nonlinear boundary velocity feedbacks \begin{document}$ f_{1}(u_{\nu{t}}) $\end{document}, \begin{document}$ f_{2}(u_{t}) $\end{document} have not polynomial growth near the origin respectively. By the combination of the Galerkin approximation, potential well method and a special basis constructed, we first obtain the global existence and uniqueness of regular solutions and weak solutions. In addition, we also investigate the explicit decay rate estimates of the energy, the ideas of which are based on the construction of a special weight function \begin{document}$ \phi(t) $\end{document} (that depends on the behaviors of the functions \begin{document}$ f_{1}(u_{\nu{t}}) $\end{document}, \begin{document}$ f_{2}(u_{t}) $\end{document} near the origin), nonlinear integral inequality and the Multiplier method.

The existence, general decay and blow-up for a plate equation with nonlinear damping and a logarithmic source term Special Issues
Gongwei Liu
2020, 28(1): 263-289 doi: 10.3934/era.2020016 +[Abstract](76) +[HTML](28) +[PDF](383.54KB)

In this paper, we consider a plate equation with nonlinear damping and logarithmic source term. By the contraction mapping principle, we establish the local existence. The global existence and decay estimate of the solution at subcritical initial energy are obtained. We also prove that the solution with negative initial energy blows up in finite time under suitable conditions. Moreover, we also give the blow-up in finite time of solution at the arbitrarily high initial energy for linear damping (i.e. \begin{document}$ m = 2 $\end{document}).

Normalized solutions for Choquard equations with general nonlinearities Special Issues
Shuai Yuan, Sitong Chen and Xianhua Tang
2020, 28(1): 291-309 doi: 10.3934/era.2020017 +[Abstract](92) +[HTML](32) +[PDF](418.12KB)

In this paper, we prove the existence of positive solutions with prescribed \begin{document}$ L^{2} $\end{document}-norm to the following Choquard equation:

where \begin{document}$ \lambda\in \mathbb{R}, \alpha\in (0,3) $\end{document} and \begin{document}$ I_{\alpha}: \mathbb{R}^3\rightarrow \mathbb{R} $\end{document} is the Riesz potential. Under the weaker conditions, by using a minimax procedure and some new analytical techniques, we show that for any \begin{document}$ c>0 $\end{document}, the above equation possesses at least a couple of weak solution \begin{document}$ (\bar{u}_c, \bar{ \lambda}_c)\in \mathcal{S}_{c}\times \mathbb{R}^- $\end{document} such that \begin{document}$ \|\bar{u}_c\|_{2}^{2} = c $\end{document}.

Long-time behavior of a class of viscoelastic plate equations Special Issues
Yang Liu
2020, 28(1): 311-326 doi: 10.3934/era.2020018 +[Abstract](76) +[HTML](34) +[PDF](371.44KB)

This paper is concerned with the initial-boundary value problem for a class of viscoelastic plate equations on an arbitrary dimensional bounded domain. Under certain assumptions on the memory kernel and the source term, the global well-posedness of solutions and the existence of global attractors are obtained.

Weak approximative compactness of hyperplane and Asplund property in Musielak-Orlicz-Bochner function spaces Special Issues
Shaoqiang Shang and Yunan Cui
2020, 28(1): 327-346 doi: 10.3934/era.2020019 +[Abstract](78) +[HTML](36) +[PDF](358.13KB)

In this paper, some criteria for weakly approximative compactness and approximative compactness of weak\begin{document}$ ^{*} $\end{document} hyperplane for Musielak-Orlicz-Bochner function spaces are given. Moreover, we also prove that, in Musielak-Orlicz-Bochner function spaces generated by strongly smooth Banach space, \begin{document}$ L_{M}^{0}(X) $\end{document} (resp \begin{document}$ L_{M}(X) $\end{document}) is an Asplund space if and only if \begin{document}$ M $\end{document} and \begin{document}$ N $\end{document} satisfy condition \begin{document}$ \Delta $\end{document}. As a corollary, we obtain that \begin{document}$ L_{M}^{0}(R) $\end{document} (resp \begin{document}$ L_{M}(R) $\end{document}) is an Asplund space if and only if \begin{document}$ M $\end{document} and \begin{document}$ N $\end{document} satisfy condition \begin{document}$ \Delta $\end{document}.

Blow-up in damped abstract nonlinear equations Special Issues
Jorge A. Esquivel-Avila
2020, 28(1): 347-367 doi: 10.3934/era.2020020 +[Abstract](99) +[HTML](40) +[PDF](404.01KB)

As a typical example of our analysis we consider a generalized Boussinesq equation, linearly damped and with a nonlinear source term. For any positive value of the initial energy, in particular for high energies, we give sufficient conditions on the initial data to conclude nonexistence of global solutions. We do our analysis in an abstract framework. We compare our results with those in the literature and we give more examples to illustrate the applicability of the abstract formulation.

Finite time blow-up and global existence of solutions for semilinear parabolic equations with nonlinear dynamical boundary condition Special Issues
Mingyou Zhang, Qingsong Zhao, Yu Liu and Wenke Li
2020, 28(1): 369-381 doi: 10.3934/era.2020021 +[Abstract](82) +[HTML](31) +[PDF](319.98KB)

For a class of semilinear parabolic equations under nonlinear dynamical boundary conditions in a bounded domain, we obtain finite time blow-up solutions when the initial data varies in the phase space \begin{document}$ H_0^1(\Omega) $\end{document} at positive initial energy level and get global solutions with the initial data at low and critical energy level. Our main tools are potential well method and concavity method.

Existence and asymptotic behavior of bound states for a class of nonautonomous Schrödinger-Poisson system Special Issues
Lirong Huang and Jianqing Chen
2020, 28(1): 383-404 doi: 10.3934/era.2020022 +[Abstract](66) +[HTML](27) +[PDF](395.33KB)

This paper is concerned with the following Schrödinger-Poisson system

where \begin{document}$ p\in (3,5) $\end{document}, \begin{document}$ K(x) $\end{document} and \begin{document}$ h(x) $\end{document} are nonnegative functions, and \begin{document}$ \mu $\end{document} is a positive parameter. Let \begin{document}$ \mu_1 > 0 $\end{document} be an isolated first eigenvalue of the eigenvalue problem \begin{document}$ -\Delta u + u = \mu h(x)u $\end{document}, \begin{document}$ u\in H^1(\mathbb{R}^3) $\end{document}. As \begin{document}$ 0<\mu\leq\mu_1 $\end{document}, we prove that \begin{document}$ (P_{\mu}) $\end{document} has at least one nonnegative bound state with positive energy. As \begin{document}$ \mu > \mu_1 $\end{document}, there is \begin{document}$ \delta > 0 $\end{document} such that for any \begin{document}$ \mu\in (\mu_1, \mu_1 + \delta) $\end{document}, \begin{document}$ (P_\mu) $\end{document} has a nonnegative ground state \begin{document}$ u_{0,\mu} $\end{document} with negative energy, and \begin{document}$ u_{0,\mu^{(n)}}\to 0 $\end{document} in \begin{document}$ H^1(\mathbb{R}^3) $\end{document} as \begin{document}$ \mu^{(n)}\downarrow \mu_1 $\end{document}. Besides, \begin{document}$ (P_\mu) $\end{document} has another nonnegative bound state \begin{document}$ u_{2,\mu} $\end{document} with positive energy, and \begin{document}$ u_{2,\mu^{(n)}}\to u_{\mu_1} $\end{document} in \begin{document}$ H^1(\mathbb{R}^3) $\end{document} as \begin{document}$ \mu^{(n)}\downarrow \mu_1 $\end{document}, where \begin{document}$ u_{\mu_1} $\end{document} is a bound state of \begin{document}$ (P_{\mu_1}) $\end{document}.

Energy minimization and preconditioning in the simulation of athermal granular materials in two dimensions Special Issues
Haolei Wang and Lei Zhang
2020, 28(1): 405-421 doi: 10.3934/era.2020023 +[Abstract](59) +[HTML](36) +[PDF](1697.67KB)

Granular materials are heterogenous grains in contact, which are ubiquitous in many scientific and engineering applications such as chemical engineering, fluid mechanics, geomechanics, pharmaceutics, and so on. Granular materials pose a great challenge to predictability, due to the presence of critical phenomena and large fluctuation of local forces. In this paper, we consider the quasi-static simulation of the dense granular media, and investigate the performances of typical minimization algorithms such as conjugate gradient methods and quasi-Newton methods. Furthermore, we develop preconditioning techniques to enhance the performance. Those methods are validated with numerical experiments for typical physically interested scenarios such as the jamming transition, the scaling law behavior close to the jamming state, and shear deformation of over jammed states.

On the mutual singularity of multifractal measures
Zied Douzi and Bilel Selmi
2020, 28(1): 423-432 doi: 10.3934/era.2020024 +[Abstract](73) +[HTML](25) +[PDF](324.42KB)

The aim of this article is to show that the multifractal Hausdorff and packing measures are mutually singular, which in particular provides an answer to Olsen's questions.

New general decay result for a fourth-order Moore-Gibson-Thompson equation with memory Special Issues
Wenjun Liu, Zhijing Chen and Zhiyu Tu
2020, 28(1): 433-457 doi: 10.3934/era.2020025 +[Abstract](98) +[HTML](34) +[PDF](421.49KB)

In this paper, we consider the fourth-order Moore-Gibson- Thompson equation with memory recently introduced by (Milan J. Math. 2017, 85: 215-234) that proposed the fourth-order model. We discuss the well-posedness of the solution by using Faedo-Galerkin method. On the other hand, for a class of relaxation functions satisfying \begin{document}$ g'(t)\leq-\xi(t)M(g(t)) $\end{document} for \begin{document}$ M $\end{document} to be increasing and convex function near the origin and \begin{document}$ \xi(t) $\end{document} to be a nonincreasing function, we establish the explicit and general energy decay result, from which we can improve the earlier related results.

The digital smash product Special Issues
Ismet Cinar, Ozgur Ege and Ismet Karaca
2020, 28(1): 459-469 doi: 10.3934/era.2020026 +[Abstract](72) +[HTML](40) +[PDF](371.39KB)

In this paper, we construct the smash product from the digital viewpoint and prove some its properties such as associativity, distributivity, and commutativity. Moreover, we present the digital suspension and the digital cone for an arbitrary digital image and give some examples of these new concepts.

Generalizations of some ordinary and extreme connectedness properties of topological spaces to relator spaces Special Issues
Muwafaq Salih and Árpád Száz
2020, 28(1): 471-548 doi: 10.3934/era.2020027 +[Abstract](112) +[HTML](169) +[PDF](771.11KB)

Motivated by some ordinary and extreme connectedness properties of topologies, we introduce several reasonable connectedness properties of relators (families of relations). Moreover, we establish some intimate connections among these properties.

More concretely, we investigate relationships among various minimalness (well-chainedness), connectedness, hyper- and ultra-connectedness, door, superset, submaximality and resolvability properties of relators.

Since most generalized topologies and all proper stacks (ascending systems) can be derived from preorder relators, the results obtained greatly extends some standard results on topologies. Moreover, they are also closely related to some former results on well-chained and connected uniformities.

Existence of best proximity points satisfying two constraint inequalities Special Issues
Duraisamy Balraj, Muthaiah Marudai, Zoran D. Mitrovic, Ozgur Ege and Veeraraghavan Piramanantham
2020, 28(1): 549-557 doi: 10.3934/era.2020028 +[Abstract](73) +[HTML](36) +[PDF](251.01KB)

In this paper, we prove the existence of best proximity point and coupled best proximity point on metric spaces with partial order for weak proximal contraction mappings such that these critical points satisfy some constraint inequalities.

On sums of four pentagonal numbers with coefficients Special Issues
Dmitry Krachun and Zhi-Wei Sun
2020, 28(1): 559-566 doi: 10.3934/era.2020029 +[Abstract](74) +[HTML](47) +[PDF](252.96KB)

The pentagonal numbers are the integers given by\begin{document}$ p_5(n) = n(3n-1)/2\ (n = 0,1,2,\ldots) $\end{document}.Let \begin{document}$ (b,c,d) $\end{document} be one of the triples \begin{document}$ (1,1,2),(1,2,3),(1,2,6) $\end{document} and \begin{document}$ (2,3,4) $\end{document}.We show that each \begin{document}$ n = 0,1,2,\ldots $\end{document} can be written as \begin{document}$ w+bx+cy+dz $\end{document} with \begin{document}$ w,x,y,z $\end{document} pentagonal numbers, which was first conjectured by Z.-W. Sun in 2016. In particular, any nonnegative integeris a sum of five pentagonal numbers two of which are equal; this refines a classical resultof Cauchy claimed by Fermat.

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