Chinese Quarterly Journal of Mathematics

Global Existence, Asympotic Behavior and Uniform Attractors for Thermoelastic Systems

LI Jiao-long, QIN Yu-ming

2017, 32(3): 221-237. doi:10.13371/j.cnki.chin.q.j.m.2017.03.001

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In this paper, we first establish the global existence and asymptotic behavior of solutions by multiplicative techniques, then further prove the existence of a uniform attractor for a thermoelastic system by using the method of uniform contractive functions. We finally investigate an alternative result of solutions for the semilinear thermoelastic systems by virtue of the semigroup method.

On the Signless Laplacian Spectral Radius of C₄-free k-cyclic Graphs

KONG Qi, WANG Li-gong

2017, 32(3): 238-245. doi:10.13371/j.cnki.chin.q.j.m.2017.03.002

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A k-cyclic graph is a connected graph of order n and size n + k-1. In this paper, we determine the maximal signless Laplacian spectral radius and the corresponding extremal graph among all C₄-free k-cyclic graphs of order n. Furthermore, we determine the first three unicycles and bicyclic, C₄-free graphs whose spectral radius of the signless Laplacian is maximal. Similar resultsare obtained for the（combinatorial） Laplacian

Global Existence and Asymptotic Behavior of Non-autonomous Timoshenko Systems

ZHENG Xiu-zhen, QIN Yu-ming

2017, 32(3): 246-254. doi:10.13371/j.cnki.chin.q.j.m.2017.03.003

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Considering Timoshenko systems under the Cattaneo thermal law, the purpose of the article is to obtain the global existence of linear Timoshenko system and semilinear Timoshenko system for the solutions of non-autonomous situations. The main method is converting the systems into abstract ODE, constructing proper spaces and proving the global existence by using semigroup methods. For asymptotic behavior, there will be a remark to describe the results.

On Rings with Finite Global Gorenstein Dimensions

REN Wei, ZHANG Yu

2017, 32(3): 255-260. doi:10.13371/j.cnki.chin.q.j.m.2017.03.004

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As a proper setting to study Gorenstein projective and injective dimensions of modules via vanishing of Gorenstein Ext-functors, a notion of a generalized Gorenstein ring is introduced, which is a non-trivial generalization of Gorenstein rings. Moreover, a new proof for Bennis and Mahdou’s equality of global Gorenstein dimension is given.

Applications of （G^’/G²）-expansion Method in Solving Nonlinear Fractional Differential Equations

KANG Zhou-zheng

2017, 32(3): 261-270. doi:10.13371/j.cnki.chin.q.j.m.2017.03.005

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In the current paper, based on fractional complex transformation, the GG2-expansion method which is used to solve differential equations of integer order is developed for finding exact solutions of nonlinear fractional differential equations with Jumarie’s modified Riemann-Liouville derivative. And then, time-fractional Burgers equation and space-fractional coupled Konopelchenko-Dubrovsky equations are provided to show that this method is effective in solving nonlinear fractional differential equations.

On the Error Term for the Number of Solutions of Certain Congruences

ZHANG Yi-feng, SHI San-ying

2017, 32(3): 271-276. doi:10.13371/j.cnki.chin.q.j.m.2017.03.006

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Let f（x） be an irreducible polynomial of degree m ≥ 2 with integer coefficients,and let r（n） denote the number of solutions x of the congruence f（x） ≡ 0（mod n） satisfying0 ≤ x < n. Define ?（x） =Σ 1≤n≤x r（n）-αx, where α is the residue of the Dedekind zeta function ζ（s, K） at its simple pole s = 1. In this paper it is shown that ∫1^X?2（x）dx?

ε{X^3-6/m+3+εif m ≥ 3,X^2+ε if m = 2,for any non-Abelian polynomial f（x） and any ε > 0. This result constitutes an improvement upon that of Lü for the error terms on average.

Convergence Rates for Elliptic Homogenization Problems in Two-dimensional Domain

ZHAO Jie, LI Hong, WANG Juan

2017, 32(3): 277-293. doi:10.13371/j.cnki.chin.q.j.m.2017.03.007

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In this paper, we study the convergence rates of solutions for second order elliptic equations with rapidly oscillating periodic coefficients in two-dimensional domain. We use an extension of the "mixed formulation" approach to obtain the representation formula satisfied by the oscillatory solution and homogenized solution by means of the particularity of solutions for equations in two-dimensional case. Then we utilize this formula in combination with the asymptotic estimates of Green or Neumann functions for operators and uniform regularity estimates of solutions to obtain convergence rates in Lp for solutions as well as gradient error estimates for Dirichlet or Neumann problems respectively.

The Representation Problems of Conjugate Spaces of l⁰(fX_ig) Type F-normed Spaces

WANG Jian-yong

2017, 32(3): 294-304. doi:10.13371/j.cnki.chin.q.j.m.2017.03.008

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Extending the results of an article published in(Acta Mathematica Sinica（2016,59（4）） by the author, for a sequence of normed spaces {X_i}, the representation problem of conjugate spaces of some l⁰（{X_i}） type F-normed spaces are studied in this paper. The algebraic representation continued equalities l⁰（{X_i}） * A=c₀₀⁰（{X_i}） * A= c₀₀（{X_i*}）,（l⁰（X））* A=（c⁰（X））* A=（c₀⁰（X））* A=(c₀₀⁰（X）)* A= c₀₀（X*）are obtained in the first part. Under weak-star topology, the topological representation c₀₀⁰（{X_i}） *, w* = c₀₀⁰（{X_i*}） is obtained in the second part. For the sequence of inner product spaces and number fields with the usual topology, the concrete forms of the basic representation theorems are obtained at last.

Global Existence, Asymptotic Behavior and Uniform Attractors for Damped Timoshenko Systems

HU Wen-song, QIN Yu-ming

2017, 32(3): 305-321. doi:10.13371/j.cnki.chin.q.j.m.2017.03.009

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In this paper, we consider the Timoshenko system as an initial-boundary value problem in a one-dimensional bounded domain. And then we establish the global existence and asymptotic behavior of solution by using the semigroup method and multiplicative techniques, then further prove the existence of a uniform attractor by using the method of uniform contractive function. The main advantage of this method is that we need only to verify compactness condition with the same type of energy estimate as that for establishing absorbing sets.

Sub-harmonic Resonance Solutions of Generalized Strongly Nonlinear Van der Pol Equation with Parametric and External Excitations

XU Dong-liang

2017, 32(3): 322-330. doi:10.13371/j.cnki.chin.q.j.m.2017.03.010

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In this paper a modified L-P method and multiple scale method are used to solve sub-harmonic resonance solutions of strong and nonlinear resonance of general Van der Pol equation with parametric and external excitations by parametric transformation. Bifurcation response equation and transition sets of sub-harmonic resonance with strong nonlinearity of general Van der Pol equation with parametric and external excitation are worked out.Besides, transition sets and bifurcation graphs are drawn to help to analysis the problems theoretically. Conclusions show that the transition sets of general and nonlinear Van der Pol equation with parametric and external excitations are more complex than those of general and nonlinear Van der Pol equation only with parametric excitation, which is helpful for the qualitative and quantitative reference for engineering and science applications.

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