Electronic Research Archive

June 2020 , Volume 28 , Issue 2

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Riemann-Liouville derivative over the space of integrable distributions
María Guadalupe Morales, Zuzana Došlá and Francisco J. Mendoza
2020, 28(2): 567-587 doi: 10.3934/era.2020030 +[Abstract](162) +[HTML](63) +[PDF](371.26KB)

In this paper, we generalize the Riemann-Liouville differential and integral operators on the space of Henstock-Kurzweil integrable distributions, \begin{document}$ D_{HK} $\end{document}. We obtain new fundamental properties of the fractional derivatives and integrals, a general version of the fundamental theorem of fractional calculus, semigroup property for the Riemann-Liouville integral operators and relations between the Riemann-Liouville integral and differential operators. Also, we achieve a generalized characterization of the solution for the Abel integral equation. Finally, we show relations for the Fourier transform of fractional derivative and integral. These results are based on the properties of the distributional Henstock-Kurzweil integral and convolution.

Proof of some conjectures involving quadratic residues
Fedor Petrov and Zhi-Wei Sun
2020, 28(2): 589-597 doi: 10.3934/era.2020031 +[Abstract](128) +[HTML](69) +[PDF](329.14KB)

We confirm several conjectures of Sun involving quadratic residues modulo odd primes. For any prime \begin{document}$ p\equiv 1\ ({\rm{mod}}\ 4) $\end{document} and integer \begin{document}$ a\not\equiv0\ ({\rm{mod}}\ p) $\end{document}, we prove that

and that

where \begin{document}$ (\frac{a}p) $\end{document} is the Legendre symbol, \begin{document}$ \varepsilon_p $\end{document} and \begin{document}$ h(p) $\end{document} are the fundamental unit and the class number of the real quadratic field \begin{document}$ \mathbb Q(\sqrt p) $\end{document} respectively, and \begin{document}$ \{x\}_p $\end{document} is the least nonnegative residue of an integer \begin{document}$ x $\end{document} modulo \begin{document}$ p $\end{document}. Also, for any prime \begin{document}$ p\equiv3\ ({\rm{mod}}\ 4) $\end{document} and \begin{document}$ {\delta} = 1, 2 $\end{document}, we determine

where \begin{document}$ T_m $\end{document} denotes the triangular number \begin{document}$ m(m+1)/2 $\end{document}.

Initial-boundary value problem for a fourth-order plate equation with Hardy-Hénon potential and polynomial nonlinearity
Xu Liu and Jun Zhou
2020, 28(2): 599-625 doi: 10.3934/era.2020032 +[Abstract](110) +[HTML](61) +[PDF](417.81KB)

In this paper, the initial-boundary value problem for a fourth-order plate equation with Hardy-Hénon potential and polynomial nonlinearity is invsitgated. First, we establish the local well-posedness of solutions by means of the semigroup theory. Then by using ordinary differential inequalities, potential well theory and energy estimate, we study the conditions on global existence and finite time blow-up. Moreover, the lifespan (i.e., the upper bound of the blow-up time) of the finite time blow-up solution is estimated.

Revisited bilinear Schrödinger estimates with applications to generalized Boussinesq equations Special Issues
Dan-Andrei Geba and Evan Witz
2020, 28(2): 627-649 doi: 10.3934/era.2020033 +[Abstract](106) +[HTML](58) +[PDF](425.86KB)

In this paper, our goal is to improve the local well-posedness theory for certain generalized Boussinesq equations by revisiting bilinear estimates related to the Schrödinger equation. Moreover, we propose a novel, automated procedure to handle the summation argument for these bounds.

Kirchhoff-type differential inclusion problems involving the fractional Laplacian and strong damping Special Issues
Mingqi Xiang, Binlin Zhang and Die Hu
2020, 28(2): 651-669 doi: 10.3934/era.2020034 +[Abstract](128) +[HTML](65) +[PDF](429.17KB)

The aim of this paper is to investigate the existence of weak solutions for a Kirchhoff-type differential inclusion wave problem involving a discontinuous set-valued term, the fractional \begin{document}$ p $\end{document}-Laplacian and linear strong damping term. The existence of weak solutions is obtained by using a regularization method combined with the Galerkin method.

Global behavior of the solutions to nonlinear Klein-Gordon equation with critical initial energy Special Issues
Milena Dimova, Natalia Kolkovska and Nikolai Kutev
2020, 28(2): 671-689 doi: 10.3934/era.2020035 +[Abstract](132) +[HTML](59) +[PDF](361.94KB)

Nonlinear Klein-Gordon equation with combined power type nonlinearity and critical initial energy is investigated. The qualitative properties of a new ordinary differential equation are studied and the concavity method of Levine is improved. Necessary and sufficient conditions for finite time blow up and global existence of the solutions are proved. New sufficient conditions on the initial data for finite time blow up, based on the necessary and sufficient ones, are obtained. The asymptotic behavior of the global solutions is also investigated.

The well-posedness and regularity of a rotating blades equation Special Issues
Lin Shen, Shu Wang and Yongxin Wang
2020, 28(2): 691-719 doi: 10.3934/era.2020036 +[Abstract](123) +[HTML](72) +[PDF](457.43KB)

In this paper, a rotating blades equation is considered. The arbitrary pre-twisted angle, arbitrary pre-setting angle and arbitrary rotating speed are taken into account when establishing the rotating blades model. The nonlinear PDEs of motion and two types of boundary conditions are derived by the extended Hamilton principle and the first-order piston theory. The well-posedness of weak solution (global in time) for the rotating blades equation with Clamped-Clamped (C-C) boundary conditions can be proved by compactness method and energy method. Strong energy estimates are derived under additional assumptions on the initial data. In addition, the existence and regularity of weak solutions (global in time) for the rotating blades equation with Clamped-Free (C-F) boundary conditions are proved as well.

Approximation by multivariate Baskakov–Kantorovich operators in Orlicz spaces
Ling-Xiong Han, Wen-Hui Li and Feng Qi
2020, 28(2): 721-738 doi: 10.3934/era.2020037 +[Abstract](116) +[HTML](51) +[PDF](375.01KB)

Utilizing some properties of multivariate Baskakov–Kantorovich operators and using \begin{document}$ K $\end{document}-functional and a decomposition technique, the authors find two equivalent theorems between the \begin{document}$ K $\end{document}-functional and modulus of smoothness, and obtain a direct theorem in the Orlicz spaces.

Certain $*$-homomorphisms acting on unital $C^{*}$-probability spaces and semicircular elements induced by $p$-adic number fields over primes $p$
Ilwoo Cho
2020, 28(2): 739-776 doi: 10.3934/era.2020038 +[Abstract](52) +[HTML](38) +[PDF](468.15KB)

In this paper, we study the Banach \begin{document}$ * $\end{document}-probability space \begin{document}$ (A\otimes_{\Bbb{C}}\Bbb{LS}, $\end{document} \begin{document}$ \tau_{A}^{0}) $\end{document} generated by a fixed unital \begin{document}$ C^{*} $\end{document}-probability space \begin{document}$ (A, $\end{document} \begin{document}$ \varphi_{A}), $\end{document} and the semicircular elements \begin{document}$ \Theta_{p,j} $\end{document} induced by \begin{document}$ p $\end{document}-adic number fields \begin{document}$ \Bbb{Q}_{p}, $\end{document} for all \begin{document}$ p $\end{document} \begin{document}$ \in $\end{document} \begin{document}$ \mathcal{P}, $\end{document} \begin{document}$ j $\end{document} \begin{document}$ \in $\end{document} \begin{document}$ \Bbb{Z}, $\end{document} where \begin{document}$ \mathcal{P} $\end{document} is the set of all primes, and \begin{document}$ \Bbb{Z} $\end{document} is the set of all integers. In particular, from the order-preserving shifts \begin{document}$ g\times h_{\pm } $\end{document} on \begin{document}$ \mathcal{P} $\end{document} \begin{document}$ \times $\end{document} \begin{document}$ \Bbb{Z}, $\end{document} and \begin{document}$ * $\end{document}-homomorphisms \begin{document}$ \theta $\end{document} on \begin{document}$ A, $\end{document} we define the corresponding \begin{document}$ * $\end{document}-homomorphisms \begin{document}$ \sigma_{(\pm ,1)}^{1:\theta } $\end{document} on \begin{document}$ A\otimes_{\Bbb{C}}\Bbb{LS}, $\end{document} and consider free-distributional data affected by them.

$ H^2 $ blowup result for a Schrödinger equation with nonlinear source term
Xuan Liu and Ting Zhang
2020, 28(2): 777-794 doi: 10.3934/era.2020039 +[Abstract](75) +[HTML](58) +[PDF](392.34KB)

In this paper, we consider the nonlinear Schrödinger equation on \begin{document}$ \mathbb{R}^N, N\ge1 $\end{document},

with \begin{document}$ H^2 $\end{document}-subcritical nonlinearities: \begin{document}$ \alpha>0, (N-4)\alpha<4 $\end{document} and Re\begin{document}$ \lambda>0 $\end{document}. For any given compact set \begin{document}$ K\subset\mathbb{R}^N $\end{document}, we construct \begin{document}$ H^2 $\end{document} solutions that are defined on \begin{document}$ (-T, 0) $\end{document} for some \begin{document}$ T>0 $\end{document}, and blow up exactly on \begin{document}$ K $\end{document} at \begin{document}$ t = 0 $\end{document}. We generalize the range of the power \begin{document}$ \alpha $\end{document} in the result of Cazenave, Han and Martel [5]. The proof is based on the energy estimates and compactness arguments.

On Seshadri constants and point-curve configurations
Marek Janasz and Piotr Pokora
2020, 28(2): 795-805 doi: 10.3934/era.2020040 +[Abstract](62) +[HTML](38) +[PDF](299.58KB)

In the note we study the multipoint Seshadri constants of \begin{document}$ \mathcal{O}_{\mathbb{P}^{2}_{\mathbb{C}}}(1) $\end{document} centered at singular loci of certain curve arrangements in the complex projective plane. Our first aim is to show that the values of Seshadri constants can be approximated with use of a combinatorial invariant which we call the configurational Seshadri constant. We study specific examples of point-curve configurations for which we provide actual values of the associated Seshadri constants. In particular, we provide an example based on Hesse point-conic configuration for which the associated Seshadri constant is computed by a line. This shows that multipoint Seshadri constants are not purely combinatorial.

A family of potential wells for a wave equation
Yang Liu and Wenke Li
2020, 28(2): 807-820 doi: 10.3934/era.2020041 +[Abstract](70) +[HTML](32) +[PDF](933.36KB)

In this paper, a family of potential wells are introduced by means of the modified depths of the potential wells. These potential wells are employed to study the initial-boundary value problem for a wave equation. The expression of the depths of the potential wells is derived. Global existence and finite time blow-up of weak solutions with the subcritical initial energy and the critical initial energy are obtained, respectively. Moreover, some numerical simulations of the depths of the potential wells are carried out.

A hybridized weak Galerkin finite element scheme for general second-order elliptic problems Special Issues
Guanrong Li, Yanping Chen and Yunqing Huang
2020, 28(2): 821-836 doi: 10.3934/era.2020042 +[Abstract](99) +[HTML](38) +[PDF](377.34KB)

In this paper, a hybridized weak Galerkin (HWG) finite element scheme is presented for solving the general second-order elliptic problems. The HWG finite element scheme is based on the use of a Lagrange multiplier defined on the element boundaries. The Lagrange multiplier provides a numerical approximation for certain derivatives of the exact solution. It is worth pointing out that a skew symmetric form has been used for handling the convection term to get the stability in the HWG formulation. Optimal order error estimates are derived for the corresponding HWG finite element approximations. A Schur complement formulation of the HWG method is introduced for implementation purpose.

Numerical simulation for 3D flow in flow channel of aeroengine turbine fan based on dimension splitting method Special Issues
Guoliang Ju, Can Chen, Rongliang Chen, Jingzhi Li, Kaitai Li and Shaohui Zhang
2020, 28(2): 837-851 doi: 10.3934/era.2020043 +[Abstract](100) +[HTML](38) +[PDF](1291.07KB)

In this paper, we introduce a dimension splitting method for simulating the air flow state of the aeroengine turbine fan. Based on the geometric model of the fan blade, the dimension splitting method establishes a semi-geodesic coordinate system. Under such coordinate system, the Navier-Stokes equations are reformulated into the combination of membrane operator equations on two-dimensional manifolds and bending operator equations along the hub circle. Using Euler central difference scheme to approximate the third variable, the new form of Navier-Stokes equations is splitting into a set of two-dimensional sub-problems. Solving these sub-problems by alternate iteration, it follows an approximate solution to Navier-Stokes equations. Furthermore, we conduct a numerical experiment to show that the dimension splitting method has a good performance by comparing with the traditional methods. Finally, we give the simulation results of the pressure and flow state of the fan blade.

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