# American Institute of Mathematical Sciences

• Previous Article
Existence of axisymmetric and homogeneous solutions of Navier-Stokes equations in cone regions
• DCDS-S Home
• This Issue
• Next Article
Global well-posedness for the Cauchy problem of generalized Boussinesq equations in the control problem regarding initial data
December  2021, 14(12): 4213-4230. doi: 10.3934/dcdss.2021134

## Nonexistence of global solutions for a class of viscoelastic wave equations

 Universidad Autónoma Metropolitana, Unidad Azcapotzalco, Av. San Pablo 180, Col. Reynosa Tamaulipas, 02200 Azcapotzalco, CDMX, México

* Corresponding author: jaea@azc.uam.mx

Received  August 2021 Revised  October 2021 Published  December 2021 Early access  November 2021

Fund Project: The author is supported by CONACYT grant 684340 and by the Universidad Autónoma Metropolitana

We consider a class of nonlinear evolution equations of second order in time, linearly damped and with a memory term. Particular cases are viscoelastic wave, Kirchhoff and Petrovsky equations. They appear in the description of the motion of deformable bodies with viscoelastic material behavior. Several articles have studied the nonexistence of global solutions of these equations due to blow-up. Most of them have considered non-positive and small positive values of the initial energy and recently some authors have analyzed these equations for any positive value of the initial energy. Within an abstract functional framework we analyze this problem and we improve the results in the literature. To this end, a new positive invariance set is introduced.

Citation: Jorge A. Esquivel-Avila. Nonexistence of global solutions for a class of viscoelastic wave equations. Discrete & Continuous Dynamical Systems - S, 2021, 14 (12) : 4213-4230. doi: 10.3934/dcdss.2021134
##### References:
 [1] J. M. Ball, Remarks on blow up and nonexistence theorems for nonlinear evolutions equations, Quart. J. Math. Oxford, 28 (1977), 473-486.  doi: 10.1093/qmath/28.4.473.  Google Scholar [2] T. Cazenave, Y. Martel and L. Zhao, Finite-time blowup for a Schrödinger equation with nonlinear source term, Discrete Contin. Dyn. Syst., 39 (2019), 1171-1183.  doi: 10.3934/dcds.2019050.  Google Scholar [3] H. Chen and H. Xu, Global existence and blow-up of solutions for infinitely degenerate semilinear pseudo-parabolic equations with logarithmic nonlinearity, Discrete Contin. Dyn. Syst., 39 (2019), 1185-1203.  doi: 10.3934/dcds.2019051.  Google Scholar [4] X. Dai, C. Yang, S. Huang, T. Yu and Y. Zhu, Finite time blow-up for a wave equation with dynamic boundary condition at critical and high energy levels in control systems, Electron. Res. Arch., 28 (2020), 91-102.  doi: 10.3934/era.2020006.  Google Scholar [5] J. A. Esquivel-Avila, Blow-up in damped abstract nonlinear equations, Electron. Res. Arch., 28 (2020), 347-367.  doi: 10.3934/era.2020020.  Google Scholar [6] M. Fabrizio and A. Morro, Mathematical Problems in Linear Viscoelasticity, SIAM Studies in Applied Mathematics, 12, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992. doi: 10.1137/1.9781611970807.  Google Scholar [7] L. Jie and L. Fei, Blow-up of solution for an integro-differential equation with arbitrary positive initial energy, Bound. Value Probl., 2015 (2015), Paper No. 96, 10 pp. doi: 10.1186/s13661-015-0361-1.  Google Scholar [8] M. Kafini and S. A. Messaoudi, A blow-up result in a nonlinear viscoelastic problem with arbitrary positive initial energy, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 20 (2013), 657-665.   Google Scholar [9] M. O. Korpusov, A. V. Ovchinnikov, A. G. Sveshnikov and E. V. Yushkov, Blow-Up in Nonlinear Equations of Mathematical Physics. Theory and Methods, De Gruyter Series in Nonlinear Analysis and Applications, 27, De Gruyter, Berlin, 2018. doi: 10.1515/9783110602074.  Google Scholar [10] H. A. Levine, Some nonexistence and instability theorems for solutions of formally parabolic equations of the form $Pu_t = -Au + \mathcal{F}(u)$, Arch. Rational Mech. Anal., 51 (1973), 371-386.  doi: 10.1007/BF00263041.  Google Scholar [11] H. A. Levine, Instability and nonexistence of global solutions to nonlinear wave equations of the form $Pu_tt = -Au + \mathcal{F}(u)$, Trans. Amer. Math. Soc., 192 (1974), 1-21.  doi: 10.2307/1996814.  Google Scholar [12] G. Li, L. Hong and W. Liu, Global nonexistence of solutions for viscoelastic wave equations of Kirchhoff type with high energy, J. Funct. Spaces Appl., 2012 (2012), Paper No. 530861, 15 pp. doi: 10.1155/2012/530861.  Google Scholar [13] G. Li, Y. Sun and W. Liu, On asymptotic behavior and blow-up of solutions for a nonlinear viscoelastic Petrovsky equation with positive initial energy, J. Funct. Spaces Appl., 2013 (2013), Paper No. 905867, 7 pp. doi: 10.1155/2013/905867.  Google Scholar [14] W. Lian and R. Xu, Global well-posedness of nonlinear wave equation with weak and strong damping terms and logarithmic source term, Adv. Nonlinear Anal., 9 (2020), 613-632.  doi: 10.1515/anona-2020-0016.  Google Scholar [15] F. Liang and H. Gao, Global existence and blow-up of solutions for a nonlinear wave equation with memory, J. Inequal. Appl., 2012 (2012), Paper No. 33, 27 pp. doi: 10.1186/1029-242X-2012-33.  Google Scholar [16] M. Liao, Q. Liu and H. Ye, Global existence and blow-up of weak solutions for a class of fractional p-Laplacian evolution equations, Adv. Nonlinear Anal., 9 (2020), 1569-1591.  doi: 10.1515/anona-2020-0066.  Google Scholar [17] Q. Lin, X. Tian, R. Xu and M. Zhang, Blow up and blow up time for degenerate Kirchhoff-type wave problems involving the fractional Laplacian with arbitrary positive initial energy, Discrete Contin. Dyn. Syst. Ser. S, 13 (2020), 2095-2107.  doi: 10.3934/dcdss.2020160.  Google Scholar [18] G. Liu, The existence, general decay and blow-up for a plate equation with nonlinear damping and a logarithmic source term, Electron. Res. Arch., 28 (2020), 263-289.  doi: 10.3934/era.2020016.  Google Scholar [19] L. Liu, F. Sun and Y. Wu, Blow-up of solutions for a nonlinear Petrovsky type equation with initial data at arbitrary high energy level, Bound. Value Probl., 2019 (2019), Paper No. 15, 18 pp. doi: 10.1186/s13661-019-1136-x.  Google Scholar [20] L. Liu, F. Sun and Y. Wu, Finite time blow-up for a nonlinear viscoelastic Petrovsky equation with high initial energy, Partial Differ. Equ. Appl., 1 (2020), Paper No. 31, 18 pp. doi: 10.1007/s42985-020-00031-1.  Google Scholar [21] Y. Liu and W. Li, A family of potential wells for a wave equation, Electron. Res. Arch., 28 (2020), 807-820.  doi: 10.3934/era.2020041.  Google Scholar [22] S. A. Messaoudi, Blow up and global existence in a nonlinear viscoelastic wave equation, Math. Nachr., 260 (2003), 58-66.  doi: 10.1002/mana.200310104.  Google Scholar [23] S. A. Messaoudi, Blow-up of positive-initial-energy solutions of a nonlinear viscoelastic hyperbolic equation, J. Math. Anal. Appl., 320 (2006), 902-915.  doi: 10.1016/j.jmaa.2005.07.022.  Google Scholar [24] H. Miyazaki, Strong blow-up instability for standing wave solutions to the system of the quadratic nonlinear Klein-Gordon equations, Discrete Contin. Dyn. Syst., 41 (2021), 2411-2445.  doi: 10.3934/dcds.2020370.  Google Scholar [25] M. Renardy, W. J. Hrusa and J. A. Nohel, Mathematical Problems in Viscoelasticity, Pitman Monographs and Surveys in Pure and Applied Mathematics, 35. Longman Scientific & Technical, Harlow; John Wiley & Sons, Inc., New York, 1987.  Google Scholar [26] T. Saanouni, Global and non global solutions for a class of coupled parabolic systems, Adv. Nonlinear Anal., 9 (2020), 1383-1401.  doi: 10.1515/anona-2020-0073.  Google Scholar [27] F. Sun, L. Liu and Y. Wu, Blow-up of solutions for a nonlinear viscoelastic wave equation with initial data at arbitrary energy level, Appl. Anal., 98 (2019), 2308-2327.  doi: 10.1080/00036811.2018.1460812.  Google Scholar [28] F. Tahamtani and M. Shahrouzi, Existence and blow up of solutions to a Petrovsky equation with memory and nonlinear source term, Bound. Value Probl., 2012 (2012), Paper No. 50, 15 pp. doi: 10.1186/1687-2770-2012-50.  Google Scholar [29] Y. Wang, A global nonexistence theorem for viscoelastic equations with arbitrary positive initial energy, Appl. Math. Lett., 22 (2009), 1394-1400.  doi: 10.1016/j.aml.2009.01.052.  Google Scholar [30] S.-T. Wu, Blow-up of solutions for an integro-differential equation with a nonlinear source, Electron. J. Differential Equations, 2006 (2006), Paper No. 45, 9 pp.  Google Scholar [31] S.-T. Wu and L.-Y. Tsai, Blow-up positive-initial-energy solutions for an integro-differential equation with nonlinear damping, Taiwanesse J. Math., 14 (2010), 2043-2058.  doi: 10.11650/twjm/1500406031.  Google Scholar [32] R. Xu, Y. Yang and Y. Liu, Global well-posedness for strongly damped viscoelastic wave equation, Appl. Anal., 92 (2013), 138-157.  doi: 10.1080/00036811.2011.601456.  Google Scholar [33] Z. Yang and G. Fan, Blow-up for the Euler-Bernoulli viscoelastic equation with a nonlinear source, Electron. J. Differential Equations, 2015 (2015), Paper No. 306, 12 pp.  Google Scholar [34] Z. Yang and Z. Gong, Blow-up solutions for viscoelastic equations of Kirchhoff type with arbitrary positive initial energy, Electron. J. Differential Equations, 2016 (2016), Paper No. 332, 8 pp.  Google Scholar [35] M. Zhang, Q. Zhao, Y. Liu and W. Li, Finite time blow-up and global existence of solutions for semilinear parabolic equations with nonlinear dynamical boundary condition, Electron. Res. Arch., 28 (2020), 369-381.  doi: 10.3934/era.2020021.  Google Scholar

show all references

##### References:
 [1] J. M. Ball, Remarks on blow up and nonexistence theorems for nonlinear evolutions equations, Quart. J. Math. Oxford, 28 (1977), 473-486.  doi: 10.1093/qmath/28.4.473.  Google Scholar [2] T. Cazenave, Y. Martel and L. Zhao, Finite-time blowup for a Schrödinger equation with nonlinear source term, Discrete Contin. Dyn. Syst., 39 (2019), 1171-1183.  doi: 10.3934/dcds.2019050.  Google Scholar [3] H. Chen and H. Xu, Global existence and blow-up of solutions for infinitely degenerate semilinear pseudo-parabolic equations with logarithmic nonlinearity, Discrete Contin. Dyn. Syst., 39 (2019), 1185-1203.  doi: 10.3934/dcds.2019051.  Google Scholar [4] X. Dai, C. Yang, S. Huang, T. Yu and Y. Zhu, Finite time blow-up for a wave equation with dynamic boundary condition at critical and high energy levels in control systems, Electron. Res. Arch., 28 (2020), 91-102.  doi: 10.3934/era.2020006.  Google Scholar [5] J. A. Esquivel-Avila, Blow-up in damped abstract nonlinear equations, Electron. Res. Arch., 28 (2020), 347-367.  doi: 10.3934/era.2020020.  Google Scholar [6] M. Fabrizio and A. Morro, Mathematical Problems in Linear Viscoelasticity, SIAM Studies in Applied Mathematics, 12, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992. doi: 10.1137/1.9781611970807.  Google Scholar [7] L. Jie and L. Fei, Blow-up of solution for an integro-differential equation with arbitrary positive initial energy, Bound. Value Probl., 2015 (2015), Paper No. 96, 10 pp. doi: 10.1186/s13661-015-0361-1.  Google Scholar [8] M. Kafini and S. A. Messaoudi, A blow-up result in a nonlinear viscoelastic problem with arbitrary positive initial energy, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 20 (2013), 657-665.   Google Scholar [9] M. O. Korpusov, A. V. Ovchinnikov, A. G. Sveshnikov and E. V. Yushkov, Blow-Up in Nonlinear Equations of Mathematical Physics. Theory and Methods, De Gruyter Series in Nonlinear Analysis and Applications, 27, De Gruyter, Berlin, 2018. doi: 10.1515/9783110602074.  Google Scholar [10] H. A. Levine, Some nonexistence and instability theorems for solutions of formally parabolic equations of the form $Pu_t = -Au + \mathcal{F}(u)$, Arch. Rational Mech. Anal., 51 (1973), 371-386.  doi: 10.1007/BF00263041.  Google Scholar [11] H. A. Levine, Instability and nonexistence of global solutions to nonlinear wave equations of the form $Pu_tt = -Au + \mathcal{F}(u)$, Trans. Amer. Math. Soc., 192 (1974), 1-21.  doi: 10.2307/1996814.  Google Scholar [12] G. Li, L. Hong and W. Liu, Global nonexistence of solutions for viscoelastic wave equations of Kirchhoff type with high energy, J. Funct. Spaces Appl., 2012 (2012), Paper No. 530861, 15 pp. doi: 10.1155/2012/530861.  Google Scholar [13] G. Li, Y. Sun and W. Liu, On asymptotic behavior and blow-up of solutions for a nonlinear viscoelastic Petrovsky equation with positive initial energy, J. Funct. Spaces Appl., 2013 (2013), Paper No. 905867, 7 pp. doi: 10.1155/2013/905867.  Google Scholar [14] W. Lian and R. Xu, Global well-posedness of nonlinear wave equation with weak and strong damping terms and logarithmic source term, Adv. Nonlinear Anal., 9 (2020), 613-632.  doi: 10.1515/anona-2020-0016.  Google Scholar [15] F. Liang and H. Gao, Global existence and blow-up of solutions for a nonlinear wave equation with memory, J. Inequal. Appl., 2012 (2012), Paper No. 33, 27 pp. doi: 10.1186/1029-242X-2012-33.  Google Scholar [16] M. Liao, Q. Liu and H. Ye, Global existence and blow-up of weak solutions for a class of fractional p-Laplacian evolution equations, Adv. Nonlinear Anal., 9 (2020), 1569-1591.  doi: 10.1515/anona-2020-0066.  Google Scholar [17] Q. Lin, X. Tian, R. Xu and M. Zhang, Blow up and blow up time for degenerate Kirchhoff-type wave problems involving the fractional Laplacian with arbitrary positive initial energy, Discrete Contin. Dyn. Syst. Ser. S, 13 (2020), 2095-2107.  doi: 10.3934/dcdss.2020160.  Google Scholar [18] G. Liu, The existence, general decay and blow-up for a plate equation with nonlinear damping and a logarithmic source term, Electron. Res. Arch., 28 (2020), 263-289.  doi: 10.3934/era.2020016.  Google Scholar [19] L. Liu, F. Sun and Y. Wu, Blow-up of solutions for a nonlinear Petrovsky type equation with initial data at arbitrary high energy level, Bound. Value Probl., 2019 (2019), Paper No. 15, 18 pp. doi: 10.1186/s13661-019-1136-x.  Google Scholar [20] L. Liu, F. Sun and Y. Wu, Finite time blow-up for a nonlinear viscoelastic Petrovsky equation with high initial energy, Partial Differ. Equ. Appl., 1 (2020), Paper No. 31, 18 pp. doi: 10.1007/s42985-020-00031-1.  Google Scholar [21] Y. Liu and W. Li, A family of potential wells for a wave equation, Electron. Res. Arch., 28 (2020), 807-820.  doi: 10.3934/era.2020041.  Google Scholar [22] S. A. Messaoudi, Blow up and global existence in a nonlinear viscoelastic wave equation, Math. Nachr., 260 (2003), 58-66.  doi: 10.1002/mana.200310104.  Google Scholar [23] S. A. Messaoudi, Blow-up of positive-initial-energy solutions of a nonlinear viscoelastic hyperbolic equation, J. Math. Anal. Appl., 320 (2006), 902-915.  doi: 10.1016/j.jmaa.2005.07.022.  Google Scholar [24] H. Miyazaki, Strong blow-up instability for standing wave solutions to the system of the quadratic nonlinear Klein-Gordon equations, Discrete Contin. Dyn. Syst., 41 (2021), 2411-2445.  doi: 10.3934/dcds.2020370.  Google Scholar [25] M. Renardy, W. J. Hrusa and J. A. Nohel, Mathematical Problems in Viscoelasticity, Pitman Monographs and Surveys in Pure and Applied Mathematics, 35. Longman Scientific & Technical, Harlow; John Wiley & Sons, Inc., New York, 1987.  Google Scholar [26] T. Saanouni, Global and non global solutions for a class of coupled parabolic systems, Adv. Nonlinear Anal., 9 (2020), 1383-1401.  doi: 10.1515/anona-2020-0073.  Google Scholar [27] F. Sun, L. Liu and Y. Wu, Blow-up of solutions for a nonlinear viscoelastic wave equation with initial data at arbitrary energy level, Appl. Anal., 98 (2019), 2308-2327.  doi: 10.1080/00036811.2018.1460812.  Google Scholar [28] F. Tahamtani and M. Shahrouzi, Existence and blow up of solutions to a Petrovsky equation with memory and nonlinear source term, Bound. Value Probl., 2012 (2012), Paper No. 50, 15 pp. doi: 10.1186/1687-2770-2012-50.  Google Scholar [29] Y. Wang, A global nonexistence theorem for viscoelastic equations with arbitrary positive initial energy, Appl. Math. Lett., 22 (2009), 1394-1400.  doi: 10.1016/j.aml.2009.01.052.  Google Scholar [30] S.-T. Wu, Blow-up of solutions for an integro-differential equation with a nonlinear source, Electron. J. Differential Equations, 2006 (2006), Paper No. 45, 9 pp.  Google Scholar [31] S.-T. Wu and L.-Y. Tsai, Blow-up positive-initial-energy solutions for an integro-differential equation with nonlinear damping, Taiwanesse J. Math., 14 (2010), 2043-2058.  doi: 10.11650/twjm/1500406031.  Google Scholar [32] R. Xu, Y. Yang and Y. Liu, Global well-posedness for strongly damped viscoelastic wave equation, Appl. Anal., 92 (2013), 138-157.  doi: 10.1080/00036811.2011.601456.  Google Scholar [33] Z. Yang and G. Fan, Blow-up for the Euler-Bernoulli viscoelastic equation with a nonlinear source, Electron. J. Differential Equations, 2015 (2015), Paper No. 306, 12 pp.  Google Scholar [34] Z. Yang and Z. Gong, Blow-up solutions for viscoelastic equations of Kirchhoff type with arbitrary positive initial energy, Electron. J. Differential Equations, 2016 (2016), Paper No. 332, 8 pp.  Google Scholar [35] M. Zhang, Q. Zhao, Y. Liu and W. Li, Finite time blow-up and global existence of solutions for semilinear parabolic equations with nonlinear dynamical boundary condition, Electron. Res. Arch., 28 (2020), 369-381.  doi: 10.3934/era.2020021.  Google Scholar
 [1] Mohammad Kafini. On the blow-up of the Cauchy problem of higher-order nonlinear viscoelastic wave equation. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021093 [2] Xiaoqiang Dai, Chao Yang, Shaobin Huang, Tao Yu, Yuanran Zhu. Finite time blow-up for a wave equation with dynamic boundary condition at critical and high energy levels in control systems. Electronic Research Archive, 2020, 28 (1) : 91-102. doi: 10.3934/era.2020006 [3] Helin Guo, Yimin Zhang, Huansong Zhou. Blow-up solutions for a Kirchhoff type elliptic equation with trapping potential. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1875-1897. doi: 10.3934/cpaa.2018089 [4] Asma Azaiez. Refined regularity for the blow-up set at non characteristic points for the vector-valued semilinear wave equation. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2397-2408. doi: 10.3934/cpaa.2019108 [5] Keng Deng, Zhihua Dong. Blow-up for the heat equation with a general memory boundary condition. Communications on Pure & Applied Analysis, 2012, 11 (5) : 2147-2156. doi: 10.3934/cpaa.2012.11.2147 [6] Yohei Fujishima. Blow-up set for a superlinear heat equation and pointedness of the initial data. Discrete & Continuous Dynamical Systems, 2014, 34 (11) : 4617-4645. doi: 10.3934/dcds.2014.34.4617 [7] István Győri, Yukihiko Nakata, Gergely Röst. Unbounded and blow-up solutions for a delay logistic equation with positive feedback. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2845-2854. doi: 10.3934/cpaa.2018134 [8] Takiko Sasaki. Convergence of a blow-up curve for a semilinear wave equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1133-1143. doi: 10.3934/dcdss.2020388 [9] Alberto Bressan, Massimo Fonte. On the blow-up for a discrete Boltzmann equation in the plane. Discrete & Continuous Dynamical Systems, 2005, 13 (1) : 1-12. doi: 10.3934/dcds.2005.13.1 [10] Yohei Fujishima. On the effect of higher order derivatives of initial data on the blow-up set for a semilinear heat equation. Communications on Pure & Applied Analysis, 2018, 17 (2) : 449-475. doi: 10.3934/cpaa.2018025 [11] Yuta Wakasugi. Blow-up of solutions to the one-dimensional semilinear wave equation with damping depending on time and space variables. Discrete & Continuous Dynamical Systems, 2014, 34 (9) : 3831-3846. doi: 10.3934/dcds.2014.34.3831 [12] Min Li, Zhaoyang Yin. Blow-up phenomena and travelling wave solutions to the periodic integrable dispersive Hunter-Saxton equation. Discrete & Continuous Dynamical Systems, 2017, 37 (12) : 6471-6485. doi: 10.3934/dcds.2017280 [13] Masahiro Ikeda, Ziheng Tu, Kyouhei Wakasa. Small data blow-up of semi-linear wave equation with scattering dissipation and time-dependent mass. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021011 [14] Jong-Shenq Guo. Blow-up behavior for a quasilinear parabolic equation with nonlinear boundary condition. Discrete & Continuous Dynamical Systems, 2007, 18 (1) : 71-84. doi: 10.3934/dcds.2007.18.71 [15] Tetsuya Ishiwata, Young Chol Yang. Numerical and mathematical analysis of blow-up problems for a stochastic differential equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 909-918. doi: 10.3934/dcdss.2020391 [16] Dapeng Du, Yifei Wu, Kaijun Zhang. On blow-up criterion for the nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems, 2016, 36 (7) : 3639-3650. doi: 10.3934/dcds.2016.36.3639 [17] Pierre Garnier. Damping to prevent the blow-up of the korteweg-de vries equation. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1455-1470. doi: 10.3934/cpaa.2017069 [18] Long Wei, Zhijun Qiao, Yang Wang, Shouming Zhou. Conserved quantities, global existence and blow-up for a generalized CH equation. Discrete & Continuous Dynamical Systems, 2017, 37 (3) : 1733-1748. doi: 10.3934/dcds.2017072 [19] Lili Du, Zheng-An Yao. Localization of blow-up points for a nonlinear nonlocal porous medium equation. Communications on Pure & Applied Analysis, 2007, 6 (1) : 183-190. doi: 10.3934/cpaa.2007.6.183 [20] Tayeb Hadj Kaddour, Michael Reissig. Blow-up results for effectively damped wave models with nonlinear memory. Communications on Pure & Applied Analysis, 2021, 20 (7&8) : 2687-2707. doi: 10.3934/cpaa.2020239

2020 Impact Factor: 2.425