September  2019, 24(9): 5121-5148. doi: 10.3934/dcdsb.2019046

Long term behavior of stochastic discrete complex Ginzburg-Landau equations with time delays in weighted spaces

1. 

School of Mathematics, Southwest Jiaotong University, Chengdu, Sichuan 610031, China

2. 

Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China

* Corresponding author: Xiaohu Wang, wangxiaohu@scu.edu.cn

Received  September 2018 Published  February 2019

Fund Project: This work was supported by NSFC (11331007, 11601446, 11701475 and 11871049) and Excellent Youth Scholars of Sichuan University (2016SCU04A15)

In this paper, we investigate the long term behavior of the solutions to a class of stochastic discrete complex Ginzburg-Landau equations with time-varying delays and driven by multiplicative white noise. We first prove the existence and uniqueness of random attractor in a weighted space for these equations. Then, we analyze the upper semicontinuity of the random attractors as the time delay approaches to zero.

Citation: Dingshi Li, Lin Shi, Xiaohu Wang. Long term behavior of stochastic discrete complex Ginzburg-Landau equations with time delays in weighted spaces. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 5121-5148. doi: 10.3934/dcdsb.2019046
References:
[1]

L. Arnold, Random Dynamical Systems, Springer-Verlag, 1998. doi: 10.1007/978-3-662-12878-7.  Google Scholar

[2]

P. W. BatesH. Lisei and K. Lu, Attractors for stochastic lattice dynamical systems, Stoch. Dynam., 6 (2006), 1-21.  doi: 10.1142/S0219493706001621.  Google Scholar

[3]

P. W. BatesK. Lu and B. Wang, Attractors of non-autonomous stochastic lattice systems in weighted spaces, Physica D, 289 (2014), 32-50.  doi: 10.1016/j.physd.2014.08.004.  Google Scholar

[4]

T. CaraballoM. J. Garrido-AtienzaB. Schmalfuss and J. Valero, Asymptotic behaviour of a stochastic semilinear dissipative functional equation without uniqueness of solutions, Discrete Contin. Dyn. Syst. - B, 14 (2010), 439-455.  doi: 10.3934/dcdsb.2010.14.439.  Google Scholar

[5]

T. CaraballoF. Morillas and J. Valero, Attractors of stochastic lattice dynamical systems with a multiplicative noise and non-Lipschitz nonlinearity, J. Differential Equations, 253 (2012), 667-693.  doi: 10.1016/j.jde.2012.03.020.  Google Scholar

[6]

T. CaraballoF. Morillas and J. Valero, On differential equations with delay in Banach spaces and attractors for retarded lattice dynamical systems, Discrete Contin. Dyn. Syst. -A, 34 (2014), 51-77.  doi: 10.3934/dcds.2014.34.51.  Google Scholar

[7]

T. ChenS. Zhou and C. Zhao, Attractors for discrete nonlinear Schrödinger equation with delay, Acta Math. Appl. Sin. Engl. Ser., 26 (2010), 633-642.  doi: 10.1007/s10255-007-7101-y.  Google Scholar

[8]

H. CuiY. Li and J. Yin, Existence and upper semicontinuity of bi-spatial pullback attractors for smoothing cocycles, Nonlinear Anal., 128 (2015), 303-324.  doi: 10.1016/j.na.2015.08.009.  Google Scholar

[9]

J. DuanK. Lu and B. Schmalfuss, Invariant manifolds for stochastic partial differential equations, Ann. Probab., 31 (2003), 2109-2135.  doi: 10.1214/aop/1068646380.  Google Scholar

[10]

A. Gu and Y. Li, Dynamic behavior of stochastic $p$-Laplacian-type lattice equations, Stoch. Dynam., 17 (2017), 1750040, 19pp. doi: 10.1142/S021949371750040X.  Google Scholar

[11]

X. Han and P. E. Kloeden, Non-autonomous lattice systems with switching effects and delayed recovery, J. Differential Equations, 261 (2016), 2986-3009.  doi: 10.1016/j.jde.2016.05.015.  Google Scholar

[12]

X. HanW. Shen and S. Zhou, Random attractors for stochastic lattice dynamical systems in weighted spaces, J. Differential Equations, 250 (2011), 1235-1266.  doi: 10.1016/j.jde.2010.10.018.  Google Scholar

[13]

X. Han, Random attractors for second order stochastic lattice dynamical systems with multiplicative noise in weighted spaces, Stoch. Dynam., 12 (2012), 1150024, 20pp. doi: 10.1142/S0219493711500249.  Google Scholar

[14]

X. Han, Asymptotic behaviors for second order stochastic lattice dynamical systems on $\mathbb Z^k$ in weighted spaces, J. Math. Anal. Appl., 397 (2013), 242-254.  doi: 10.1016/j.jmaa.2012.07.015.  Google Scholar

[15]

N. KarachaliosH. Nistazakis and A. Yannacopoulos, Asymptotic behavior of solutions of complex discrete evolution equations: the discrete Ginzburg-Landau equation, Discrete Contin. Dyn. Syst -A, 19 (2007), 711-736.  doi: 10.3934/dcds.2007.19.711.  Google Scholar

[16]

D. Li and L. Shi, Upper semicontinuity of random attractors of stochastic discrete complex Ginzburg-Landau equations with time-varying delays in the delay, J. Difference Equ. Appl., 24 (2018), 872-897.  doi: 10.1080/10236198.2018.1437913.  Google Scholar

[17]

X. LiK. Wei and H. Zhang, Exponential attractors for lattice dynamical systems in weighted spaces, Acta Appl. Math., 114 (2011), 157-172.  doi: 10.1007/s10440-011-9606-x.  Google Scholar

[18]

V. A. Liskevich and M. A. Perelmuter, Analyticity of submarkovian semigroups, Proc. Amer. Math. Soc., 123 (1995), 1097-1104.  doi: 10.2307/2160706.  Google Scholar

[19]

Y. Lv and J. Sun, Asymptotic behavior of stochastic discrete complex Ginzburg-Landau equations, Physica D, 221 (2006), 157-169.  doi: 10.1016/j.physd.2006.07.023.  Google Scholar

[20]

J. Shu, Random attractors for stochastic discrete Klein-Gordon-Schrödinger equations driven by fractional Brownian motions, Discrete Contin. Dyn. Syst. - B, 22 (2017), 1587-1599.  doi: 10.3934/dcdsb.2017077.  Google Scholar

[21]

B. Wang, Suffcient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Differential Equations, 253 (2012), 1544-1583.  doi: 10.1016/j.jde.2012.05.015.  Google Scholar

[22]

B. Wang, Uniform attractors of nonautonomous discrete reaction-diffusion systems in weighted spaces, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 18 (2008), 695-716.  doi: 10.1142/S0218127408020598.  Google Scholar

[23]

Y. Wang and K. Bai, Pullback attractors for a class of nonlinear lattices with delays, Discrete Contin. Dyn. Syst. - B, 20 (2015), 1213-1230.  doi: 10.3934/dcdsb.2015.20.1213.  Google Scholar

[24]

P. Wang, Y. Huang and X. Wang, Random attractors for stochastic discrete complex non-autonomous Ginzburg-Landau equations with multiplicative noise, Adv. Difference Equ., 2015 (2015), 15pp. doi: 10.1186/s13662-015-0575-7.  Google Scholar

[25]

X. WangK. Lu and B. Wang, Long term behavior of delay parabolic equations with additive noise and deterministic time dependent forcing, SIAM J. Appl. Dynam. Syst., 14 (2015), 1018-1047.  doi: 10.1137/140991819.  Google Scholar

[26]

X. WangK. Lu and B. Wang, Exponential stability of non-autonomous stochastic delay lattice systems driven by a multiplicative white noise, J. Dynam. Differential Equations, 28 (2016), 1309-1335.  doi: 10.1007/s10884-015-9448-8.  Google Scholar

[27]

X. WangS. Li and D. Xu, Random attractors for second-order stochastic lattice dynamical systems, Nonlinear Anal., 72 (2010), 483-494.  doi: 10.1016/j.na.2009.06.094.  Google Scholar

[28]

Z. Wang and S. Zhou, Existence and upper semicontinuity of attractors for non-autonomous stochastic lattice systems with random coupled coefficients, Commun. Pure Appl. Anal., 15 (2016), 2221-2245.  doi: 10.3934/cpaa.2016035.  Google Scholar

[29]

Z. Wang and S. Zhou, Random attractors for stochastic retarded lattice systems, J. Difference Equations Appl., 19 (2013), 1523-1543.  doi: 10.1080/10236198.2013.765412.  Google Scholar

[30]

W. Yan, Y. Li and S. Ji, Random attractors for first order stochastic retarded lattice dynamical systems, J. Math. Phys., 51 (2010), 032702, 17pp. doi: 10.1063/1.3319566.  Google Scholar

[31]

D. Yang, The asymptotic behavior of the stochastic Ginzburg-Landau equation with multiplicative noise, J. Math. Phys., 45 (2004), 4064-4076.  doi: 10.1063/1.1794365.  Google Scholar

[32]

C. Zhang and L. Zhao, The attractors for 2nd-order stochastic delay lattice systems, Discrete Contin. Dyn. Syst. - A, 37 (2017), 575-590.  doi: 10.3934/dcds.2017023.  Google Scholar

[33]

M. Zhao and S. Zhou, Random attractor of non-autonomous stochastic Boussinesq lattice system, J. Math. Phys., 56 (2015), 092702, 16pp. doi: 10.1063/1.4930195.  Google Scholar

[34]

C. Zhao and S. Zhou, Limit behavior of global attractors for the complex Ginzburg-Landau equation on infinite lattices, Appl. Math. Lett., 21 (2008), 628-635.  doi: 10.1016/j.aml.2007.07.016.  Google Scholar

[35]

C. Zhao and S. Zhou, Attractors of retarded first order lattice systems, Nonlinearity, 20 (2007), 1987-2006.  doi: 10.1088/0951-7715/20/8/010.  Google Scholar

[36]

S. Zhou and Z. Wang, Random attractors for stochastic retarded lattice systems, J. Difference Equ. Appl., 19 (2013), 1523-1543.  doi: 10.1080/10236198.2013.765412.  Google Scholar

show all references

References:
[1]

L. Arnold, Random Dynamical Systems, Springer-Verlag, 1998. doi: 10.1007/978-3-662-12878-7.  Google Scholar

[2]

P. W. BatesH. Lisei and K. Lu, Attractors for stochastic lattice dynamical systems, Stoch. Dynam., 6 (2006), 1-21.  doi: 10.1142/S0219493706001621.  Google Scholar

[3]

P. W. BatesK. Lu and B. Wang, Attractors of non-autonomous stochastic lattice systems in weighted spaces, Physica D, 289 (2014), 32-50.  doi: 10.1016/j.physd.2014.08.004.  Google Scholar

[4]

T. CaraballoM. J. Garrido-AtienzaB. Schmalfuss and J. Valero, Asymptotic behaviour of a stochastic semilinear dissipative functional equation without uniqueness of solutions, Discrete Contin. Dyn. Syst. - B, 14 (2010), 439-455.  doi: 10.3934/dcdsb.2010.14.439.  Google Scholar

[5]

T. CaraballoF. Morillas and J. Valero, Attractors of stochastic lattice dynamical systems with a multiplicative noise and non-Lipschitz nonlinearity, J. Differential Equations, 253 (2012), 667-693.  doi: 10.1016/j.jde.2012.03.020.  Google Scholar

[6]

T. CaraballoF. Morillas and J. Valero, On differential equations with delay in Banach spaces and attractors for retarded lattice dynamical systems, Discrete Contin. Dyn. Syst. -A, 34 (2014), 51-77.  doi: 10.3934/dcds.2014.34.51.  Google Scholar

[7]

T. ChenS. Zhou and C. Zhao, Attractors for discrete nonlinear Schrödinger equation with delay, Acta Math. Appl. Sin. Engl. Ser., 26 (2010), 633-642.  doi: 10.1007/s10255-007-7101-y.  Google Scholar

[8]

H. CuiY. Li and J. Yin, Existence and upper semicontinuity of bi-spatial pullback attractors for smoothing cocycles, Nonlinear Anal., 128 (2015), 303-324.  doi: 10.1016/j.na.2015.08.009.  Google Scholar

[9]

J. DuanK. Lu and B. Schmalfuss, Invariant manifolds for stochastic partial differential equations, Ann. Probab., 31 (2003), 2109-2135.  doi: 10.1214/aop/1068646380.  Google Scholar

[10]

A. Gu and Y. Li, Dynamic behavior of stochastic $p$-Laplacian-type lattice equations, Stoch. Dynam., 17 (2017), 1750040, 19pp. doi: 10.1142/S021949371750040X.  Google Scholar

[11]

X. Han and P. E. Kloeden, Non-autonomous lattice systems with switching effects and delayed recovery, J. Differential Equations, 261 (2016), 2986-3009.  doi: 10.1016/j.jde.2016.05.015.  Google Scholar

[12]

X. HanW. Shen and S. Zhou, Random attractors for stochastic lattice dynamical systems in weighted spaces, J. Differential Equations, 250 (2011), 1235-1266.  doi: 10.1016/j.jde.2010.10.018.  Google Scholar

[13]

X. Han, Random attractors for second order stochastic lattice dynamical systems with multiplicative noise in weighted spaces, Stoch. Dynam., 12 (2012), 1150024, 20pp. doi: 10.1142/S0219493711500249.  Google Scholar

[14]

X. Han, Asymptotic behaviors for second order stochastic lattice dynamical systems on $\mathbb Z^k$ in weighted spaces, J. Math. Anal. Appl., 397 (2013), 242-254.  doi: 10.1016/j.jmaa.2012.07.015.  Google Scholar

[15]

N. KarachaliosH. Nistazakis and A. Yannacopoulos, Asymptotic behavior of solutions of complex discrete evolution equations: the discrete Ginzburg-Landau equation, Discrete Contin. Dyn. Syst -A, 19 (2007), 711-736.  doi: 10.3934/dcds.2007.19.711.  Google Scholar

[16]

D. Li and L. Shi, Upper semicontinuity of random attractors of stochastic discrete complex Ginzburg-Landau equations with time-varying delays in the delay, J. Difference Equ. Appl., 24 (2018), 872-897.  doi: 10.1080/10236198.2018.1437913.  Google Scholar

[17]

X. LiK. Wei and H. Zhang, Exponential attractors for lattice dynamical systems in weighted spaces, Acta Appl. Math., 114 (2011), 157-172.  doi: 10.1007/s10440-011-9606-x.  Google Scholar

[18]

V. A. Liskevich and M. A. Perelmuter, Analyticity of submarkovian semigroups, Proc. Amer. Math. Soc., 123 (1995), 1097-1104.  doi: 10.2307/2160706.  Google Scholar

[19]

Y. Lv and J. Sun, Asymptotic behavior of stochastic discrete complex Ginzburg-Landau equations, Physica D, 221 (2006), 157-169.  doi: 10.1016/j.physd.2006.07.023.  Google Scholar

[20]

J. Shu, Random attractors for stochastic discrete Klein-Gordon-Schrödinger equations driven by fractional Brownian motions, Discrete Contin. Dyn. Syst. - B, 22 (2017), 1587-1599.  doi: 10.3934/dcdsb.2017077.  Google Scholar

[21]

B. Wang, Suffcient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Differential Equations, 253 (2012), 1544-1583.  doi: 10.1016/j.jde.2012.05.015.  Google Scholar

[22]

B. Wang, Uniform attractors of nonautonomous discrete reaction-diffusion systems in weighted spaces, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 18 (2008), 695-716.  doi: 10.1142/S0218127408020598.  Google Scholar

[23]

Y. Wang and K. Bai, Pullback attractors for a class of nonlinear lattices with delays, Discrete Contin. Dyn. Syst. - B, 20 (2015), 1213-1230.  doi: 10.3934/dcdsb.2015.20.1213.  Google Scholar

[24]

P. Wang, Y. Huang and X. Wang, Random attractors for stochastic discrete complex non-autonomous Ginzburg-Landau equations with multiplicative noise, Adv. Difference Equ., 2015 (2015), 15pp. doi: 10.1186/s13662-015-0575-7.  Google Scholar

[25]

X. WangK. Lu and B. Wang, Long term behavior of delay parabolic equations with additive noise and deterministic time dependent forcing, SIAM J. Appl. Dynam. Syst., 14 (2015), 1018-1047.  doi: 10.1137/140991819.  Google Scholar

[26]

X. WangK. Lu and B. Wang, Exponential stability of non-autonomous stochastic delay lattice systems driven by a multiplicative white noise, J. Dynam. Differential Equations, 28 (2016), 1309-1335.  doi: 10.1007/s10884-015-9448-8.  Google Scholar

[27]

X. WangS. Li and D. Xu, Random attractors for second-order stochastic lattice dynamical systems, Nonlinear Anal., 72 (2010), 483-494.  doi: 10.1016/j.na.2009.06.094.  Google Scholar

[28]

Z. Wang and S. Zhou, Existence and upper semicontinuity of attractors for non-autonomous stochastic lattice systems with random coupled coefficients, Commun. Pure Appl. Anal., 15 (2016), 2221-2245.  doi: 10.3934/cpaa.2016035.  Google Scholar

[29]

Z. Wang and S. Zhou, Random attractors for stochastic retarded lattice systems, J. Difference Equations Appl., 19 (2013), 1523-1543.  doi: 10.1080/10236198.2013.765412.  Google Scholar

[30]

W. Yan, Y. Li and S. Ji, Random attractors for first order stochastic retarded lattice dynamical systems, J. Math. Phys., 51 (2010), 032702, 17pp. doi: 10.1063/1.3319566.  Google Scholar

[31]

D. Yang, The asymptotic behavior of the stochastic Ginzburg-Landau equation with multiplicative noise, J. Math. Phys., 45 (2004), 4064-4076.  doi: 10.1063/1.1794365.  Google Scholar

[32]

C. Zhang and L. Zhao, The attractors for 2nd-order stochastic delay lattice systems, Discrete Contin. Dyn. Syst. - A, 37 (2017), 575-590.  doi: 10.3934/dcds.2017023.  Google Scholar

[33]

M. Zhao and S. Zhou, Random attractor of non-autonomous stochastic Boussinesq lattice system, J. Math. Phys., 56 (2015), 092702, 16pp. doi: 10.1063/1.4930195.  Google Scholar

[34]

C. Zhao and S. Zhou, Limit behavior of global attractors for the complex Ginzburg-Landau equation on infinite lattices, Appl. Math. Lett., 21 (2008), 628-635.  doi: 10.1016/j.aml.2007.07.016.  Google Scholar

[35]

C. Zhao and S. Zhou, Attractors of retarded first order lattice systems, Nonlinearity, 20 (2007), 1987-2006.  doi: 10.1088/0951-7715/20/8/010.  Google Scholar

[36]

S. Zhou and Z. Wang, Random attractors for stochastic retarded lattice systems, J. Difference Equ. Appl., 19 (2013), 1523-1543.  doi: 10.1080/10236198.2013.765412.  Google Scholar

[1]

N. I. Karachalios, Hector E. Nistazakis, Athanasios N. Yannacopoulos. Asymptotic behavior of solutions of complex discrete evolution equations: The discrete Ginzburg-Landau equation. Discrete & Continuous Dynamical Systems - A, 2007, 19 (4) : 711-736. doi: 10.3934/dcds.2007.19.711

[2]

Michael Stich, Carsten Beta. Standing waves in a complex Ginzburg-Landau equation with time-delay feedback. Conference Publications, 2011, 2011 (Special) : 1329-1334. doi: 10.3934/proc.2011.2011.1329

[3]

Yuta Kugo, Motohiro Sobajima, Toshiyuki Suzuki, Tomomi Yokota, Kentarou Yoshii. Solvability of a class of complex Ginzburg-Landau equations in periodic Sobolev spaces. Conference Publications, 2015, 2015 (special) : 754-763. doi: 10.3934/proc.2015.0754

[4]

N. I. Karachalios, H. E. Nistazakis, A. N. Yannacopoulos. Remarks on the asymptotic behavior of solutions of complex discrete Ginzburg-Landau equations. Conference Publications, 2005, 2005 (Special) : 476-486. doi: 10.3934/proc.2005.2005.476

[5]

Hans G. Kaper, Peter Takáč. Bifurcating vortex solutions of the complex Ginzburg-Landau equation. Discrete & Continuous Dynamical Systems - A, 1999, 5 (4) : 871-880. doi: 10.3934/dcds.1999.5.871

[6]

Noboru Okazawa, Tomomi Yokota. Subdifferential operator approach to strong wellposedness of the complex Ginzburg-Landau equation. Discrete & Continuous Dynamical Systems - A, 2010, 28 (1) : 311-341. doi: 10.3934/dcds.2010.28.311

[7]

Sen-Zhong Huang, Peter Takáč. Global smooth solutions of the complex Ginzburg-Landau equation and their dynamical properties. Discrete & Continuous Dynamical Systems - A, 1999, 5 (4) : 825-848. doi: 10.3934/dcds.1999.5.825

[8]

Hongzi Cong, Jianjun Liu, Xiaoping Yuan. Quasi-periodic solutions for complex Ginzburg-Landau equation of nonlinearity $|u|^{2p}u$. Discrete & Continuous Dynamical Systems - S, 2010, 3 (4) : 579-600. doi: 10.3934/dcdss.2010.3.579

[9]

Yueling Jia, Zhaohui Huo. Inviscid limit behavior of solution for the multi-dimensional derivative complex Ginzburg-Landau equation. Kinetic & Related Models, 2014, 7 (1) : 57-77. doi: 10.3934/krm.2014.7.57

[10]

Hong Lu, Shujuan Lü, Mingji Zhang. Fourier spectral approximations to the dynamics of 3D fractional complex Ginzburg-Landau equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2539-2564. doi: 10.3934/dcds.2017109

[11]

Qiongwei Huang, Jiashi Tang. Bifurcation of a limit cycle in the ac-driven complex Ginzburg-Landau equation. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 129-141. doi: 10.3934/dcdsb.2010.14.129

[12]

Bo You, Yanren Hou, Fang Li, Jinping Jiang. Pullback attractors for the non-autonomous quasi-linear complex Ginzburg-Landau equation with $p$-Laplacian. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1801-1814. doi: 10.3934/dcdsb.2014.19.1801

[13]

Feng Zhou, Chunyou Sun. Dynamics for the complex Ginzburg-Landau equation on non-cylindrical domains I: The diffeomorphism case. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3767-3792. doi: 10.3934/dcdsb.2016120

[14]

Noboru Okazawa, Tomomi Yokota. Smoothing effect for generalized complex Ginzburg-Landau equations in unbounded domains. Conference Publications, 2001, 2001 (Special) : 280-288. doi: 10.3934/proc.2001.2001.280

[15]

N. Maaroufi. Topological entropy by unit length for the Ginzburg-Landau equation on the line. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 647-662. doi: 10.3934/dcds.2014.34.647

[16]

Jingna Li, Li Xia. The Fractional Ginzburg-Landau equation with distributional initial data. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2173-2187. doi: 10.3934/cpaa.2013.12.2173

[17]

Satoshi Kosugi, Yoshihisa Morita, Shoji Yotsutani. A complete bifurcation diagram of the Ginzburg-Landau equation with periodic boundary conditions. Communications on Pure & Applied Analysis, 2005, 4 (3) : 665-682. doi: 10.3934/cpaa.2005.4.665

[18]

Jun Yang. Vortex structures for Klein-Gordon equation with Ginzburg-Landau nonlinearity. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 2359-2388. doi: 10.3934/dcds.2014.34.2359

[19]

Dingshi Li, Xiaohu Wang. Asymptotic behavior of stochastic complex Ginzburg-Landau equations with deterministic non-autonomous forcing on thin domains. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 449-465. doi: 10.3934/dcdsb.2018181

[20]

Jungho Park. Bifurcation and stability of the generalized complex Ginzburg--Landau equation. Communications on Pure & Applied Analysis, 2008, 7 (5) : 1237-1253. doi: 10.3934/cpaa.2008.7.1237

2018 Impact Factor: 1.008

Metrics

  • PDF downloads (86)
  • HTML views (366)
  • Cited by (0)

Other articles
by authors

[Back to Top]