Table of Content

  

20 June 2021
Volume 56 Issue 6

Alternative results of a class of quasilinear transient parabolic equations

LI Yuan-fei, LI Dan-dan, CHEN Xue-jiao, SHI Jin-cheng

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2021, 56(6):  1-9.  doi:10.6040/j.issn.1671-9352.0.2020.687

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In this paper, the spatial Phragmén-Lindelöf type alternative of solutions for quasilinear parabolic systems satisfying nonlinear dynamic conditions on the side of the cylinder is considered. By using the technique of differential inequality, it is proved that the solution increases exponentially or decays exponentially with spatial variables. By setting an arbitrary positive constant, more accurate decay rate and growth rate are obtained. Finally, the alternative theorem is extended to the heat equation in binary mixtures.

Existence of IS-asymptotically periodic mild solutions for a class of impulsive evolution equations

YUAN Tian-jiao, LI Qiang

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2021, 56(6):  10-21.  doi:10.6040/j.issn.1671-9352.0.2020.571

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In this paper, a class of evolution equations with non-instantaneous impulses in a Banach space X is considered. Under the assumption that the upper and lower solutions of the evolution equation with non-instantaneous impulses exist, a monotone iterative method is constructed and the existence and uniqueness of IS-asymptotically ω-periodic mild solutions are obtained. Finally, the application of the main results in partial differential equations is given.

Time-dependent attractors of the wave equations with strong damping on Rⁿ

WU Xiao-xia, MA Qiao-zhen

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2021, 56(6):  22-29.  doi:10.6040/j.issn.1671-9352.0.2020.359

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Based on the tail estimation technique and condition (C_t), the asymptotic compactness of the process related to the problem is obtained, and the difficulty of Sobolev embedding noncompactness and Poincaré inequality is not established in the entire space is overcome, thus the existence of time-dependent attractors for the wave equations with strong damping and decay coefficients over unbounded domains is proved.

Pullback attractor for the non-autonomous Berger equation with nonlinear damping

DU Ya-li, WANG Xuan

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2021, 56(6):  30-41.  doi:10.6040/j.issn.1671-9352.0.2020.656

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This paper investigates the asymptotic of solution for the Berger equation with nonlinear damping and time-dependent external force. Then the existence time-dependent pullback attractor is proved by use of contractive function and asymptotic priori estimation.

Existence of solution for the critical biharmonic equations involving Rellich potentials

JIANG Rui-ting, ZHAI Cheng-bo

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2021, 56(6):  42-46.  doi:10.6040/j.issn.1671-9352.0.2020.542

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This paper considers a class of critical biharmonic equations involving Rellich potentials Δ²u-μu/(|x|4)=(|u|2^*(s)-2u)/(|x|^s)+λf(x,u)in a bounded domain. By mountain pass theorem, the existence of at least a nontrivial solution is obtained.

Gradient estimates for weak solutions of A-harmonic equation under nonstandard growth

ZHOU Yan-xia, WANG Xin-ru, XU Xiu-juan

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2021, 56(6):  47-55.  doi:10.6040/j.issn.1671-9352.0.2020.563

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By establishing the reverse Hölder inequality of SymbolQC@u and using the method of maximal function, the gradient estimation of the weak solution for the non-homogeneous A-harmonic equation div A(x,SymbolQC@u)=B(x,SymbolQC@u) is obtained.

Solution to the fractional Schrödinger-Poisson systems with critical term

GUO Kai-li, FENG Xiao-jing

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2021, 56(6):  56-63.  doi:10.6040/j.issn.1671-9352.0.2020.713

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This paper studies a class of fractional Schrödinger-Poisson systems with critical term, which has recently been widely used in optimization, finance, reaction diffusion and so on. Since the problem in the system has two critical terms, it is difficult to estimate the critical value of mountain pass; and the potential function is neither periodic nor asymptotic periodic, the usual concentration-compactness method is invalid. So we employ variational method and modified concentration-compactness principle to obtain the existence of nontrivial solution of this system. This result supplements and expands on the previous results on fractional Schrödinger-Poisson systems.

Identification of source term for fractional diffusion-wave equation with Neumann boundary conditions

QI Bin, CHENG Hao

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2021, 56(6):  64-73.  doi:10.6040/j.issn.1671-9352.0.2020.582

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The source term identification of the time-fractional diffusion-wave equation with Neumann boundary conditions is studied. An improved iterative regularization method is constructed to calculate the regularization solution of the source term. The error estimates between the regularization solution and the exact solution are given under the prior and the posterior regularization parameter choice rules. Numerical examples verify the effectiveness of the iterative regularization method.

Quasi-boundary value regularization method for inhomogeneous sideways heat equation

WANG Feng-xia, XIONG Xiang-tuan

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2021, 56(6):  74-80.  doi:10.6040/j.issn.1671-9352.0.2020.539

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In the inhomogeneous sideways heat equation, it is assumed that the heat source is largely dependent on space and time and cannot be ignored. Since the solution of the problem(if it exists)is discontinuously dependent on the data, this is a typical ill-posed problem, and most of the literature only deals with homogeneous sideways heat equation. By using Fourier transform and quasi-boundary value regularization method, the inhomogeneous sideways heat equation is studied, and the stable approximate solution is obtained. The error estimate of the stability is given under the prior parameter selection and the posterior parameter selection rule.

Progressive-iterative approximation by the triangular β-B curves with shape parameter

WANG Zeng-zhen, LIU Hua-yong, ZHA Dong-dong

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2021, 56(6):  81-94.  doi:10.6040/j.issn.1671-9352.0.2021.136

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In order to accelerate the convergence speed of the progressive-iterative approximation method and overcome the defect that the general B-spline curves cannot represent curves such as circles or ellipses, this paper discusses its(weighted)progressive-iterative approximation based on β-B curves. Compared with general cubic uniform polynomial B-spline curve, the former has better continuity. The iterative matrix of the(weighted)progressive-iterative approximation is obtained according to the selected β-B basis function. Based on the conclusion that the spectral radius is the smallest and the convergence speed is the fastest,the optimal shape parameters of the(weighted)progressive-iterative approximation and the optimal weight w of the weighted progressive-pterative approximation method are derived. Then this paper does convergence analysis on them respectively. Finally, numerical examples are given to analyze the iteration speed and iteration error when the shape parameters are different. The experimental results show that the convergence speed is the fastest when the shape parameter and weight are optimal.

Solvability conditions for a class of tensor inverse eigenvalue problems

DAI Li-fang, LIANG Mao-lin, RAN Yan-ping

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2021, 56(6):  95-102.  doi:10.6040/j.issn.1671-9352.0.2020.463

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Using the properties of Moore-Penrose generalized inverses of tensors, the solvability conditions for the existence of the solution to the inverse eigenvalue problem of Hermitian tensors with Einstein product, as well as the general solution, are obtained. Meanwhile, the unique solution to the associated tensor approximation problem for any given tensor is given. The performed numerical results illustrate the feasibility of these results.

Solutions of the quaternion matrix equation based on semi-tensor product of matrices

DING Wen-xu, LI Ying, WANG Dong, ZHAO Jian-li

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2021, 56(6):  103-110.  doi:10.6040/j.issn.1671-9352.0.2020.682

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A new kind of real vector representation of quaternion matrix is proposed which is applied to study the Hermitian or anti-Hermitian solution of the quaternion matrix equation. Combined this real vector representation with semi-tensor product of matrics, the problem of quaternion matrix equation is transformed into the corresponding problem in real number field. And then, by studying the structural characteristics of quaternion Hermitian matrix and anti-Hermitian matrix, the useful information in the real vector representation is extracted and the redundancy is removed, then the original problem is simplified by reducing the dimension. This method can be applied to similar problems with different constraint.