American Institute of Mathematical Sciences

February  2019, 39(2): 1185-1203. doi: 10.3934/dcds.2019051

Global existence and blow-up of solutions for infinitely degenerate semilinear pseudo-parabolic equations with logarithmic nonlinearity

 School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China

* Corresponding author: Hua Chen

Received  June 2018 Revised  August 2018 Published  November 2018

Fund Project: This work is supported by the NSFC under the grant 11631011

In this paper, we study the initial-boundary value problem for infinitely degenerate semilinear pseudo-parabolic equations with logarithmic nonlinearity $u_t-\triangle_{X} u_t-\triangle_{X} u=u\log|u|$, where $X=(X_1, X_2, ··· , X_m)$ is an infinitely degenerate system of vector fields, and $\triangle_{X}:=\sum^{m}_{j=1}X^{2}_{j}$ is an infinitely degenerate elliptic operator. Using potential well method, we first prove the invariance of some sets and vacuum isolating of solutions. Then, by the Galerkin approximation technique, the logarithmic Sobolev inequality and Poincaré inequality, we obtain the global existence and blow-up at $+∞$ of solutions with low initial energy or critical initial energy, and discuss the asymptotic behavior of the solutions.

Citation: Hua Chen, Huiyang Xu. Global existence and blow-up of solutions for infinitely degenerate semilinear pseudo-parabolic equations with logarithmic nonlinearity. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 1185-1203. doi: 10.3934/dcds.2019051
References:
 [1] J. M. Ball, Remarks on blow-up and nonexistence theorems for nonlinear evolution equations, Quart. J. Math. Oxford Ser., 28 (1977), 473-486. doi: 10.1093/qmath/28.4.473. Google Scholar [2] G. Barenblat, I. Zheltov and I. Kochiva, Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks, J. Appl. Math. Mech., 24 (1960), 1286-1303. Google Scholar [3] E. D. Benedetto and M. Pierre, On the maximum principle for pseudoparabolic equations, Indiana Univ. Math. J., 30 (1981), 821-854. doi: 10.1512/iumj.1981.30.30062. Google Scholar [4] T. B. Benjamin, J. L. Bona and J. J. Mahony, Model equations for long waves in nonlinear dispersive systems, Philos. Trans. R. Soc. Lond. Ser. A, 272 (1972), 47-78. doi: 10.1098/rsta.1972.0032. Google Scholar [5] H. Brill, A semilinear Sobolev evolution equation in a Banach space, J. Differential Equations, 24 (1977), 412-425. doi: 10.1016/0022-0396(77)90009-2. Google Scholar [6] Y. Cao, J. Yin and C. Wang, Cauchy problems of semilinear pseudo-parabolic equations, J. Differential Equations, 246 (2009), 4568-4590. doi: 10.1016/j.jde.2009.03.021. Google Scholar [7] H. Chen and K. Li, The existence and regularity of multiple solutions for a class of infinitely degenerate elliptic equations, Math Nach, 282 (2009), 368-385. doi: 10.1002/mana.200710742. Google Scholar [8] H. Chen, P. Luo and G. Liu, Global solution and blow up of a semilinear heat equation with logarithmic nonlinearity, J. Math. Anal. Appl., 422 (2015), 84-98. doi: 10.1016/j.jmaa.2014.08.030. Google Scholar [9] H. Chen, P. Luo and S. Tian, Existence and regularity of multiple solutions forinfinitely degenerate nonlinear elliptic equations with singular potential, J. Differ Equ., 257 (2014), 3300-3333. doi: 10.1016/j.jde.2014.06.014. Google Scholar [10] H. Chen, P. Luo and S. Tian, Multiplicity and regularity of solutions for infinitely degenerate elliptic equations with a free perturbation, J. Math. Pures Appl., 103 (2015), 849-867. doi: 10.1016/j.matpur.2014.09.004. Google Scholar [11] H. Chen and S. Tian, Initial boundary value problem for a class of semilinear pseudo-parabolic equations with logarithmic nonlinearity, J. Differential Equations, 285 (2015), 4424-4442. doi: 10.1016/j.jde.2015.01.038. Google Scholar [12] M. Christ, Hypoellipticity in the infinitely degenerate regime, Complex Analysis and Geometry (Columbus, OH, 1999), 59-84, Ohio State Univ. Math. Res. Inst. Publ., 9, de Gruyter, Berlin, 2001. Google Scholar [13] C. David and W. Jet, Asymptotic behaviour of the fundamental solution to the equation of heat conduction in two temperatures, J. Math. Anal. Appl., 69 (1979), 411-418. doi: 10.1016/0022-247X(79)90152-5. Google Scholar [14] V. R. Gopala Rao and T. W. Ting, Solutions of pseudo-heat equations in the whole space, Arch. Ration. Mech. Anal., 49 (1972/73), 57-78. doi: 10.1007/BF00281474. Google Scholar [15] L. Hörmander, Hypoelliptic second order differential equations, Acta Math., 119 (1967), 147-171. doi: 10.1007/BF02392081. Google Scholar [16] S. Ji, J. Yin and Y. Cao, Instability of positive periodic solutions for semilinear pseudo-parabolic equations with logarithmic nonlinearity, J. Differential Equations, 261 (2016), 5446-5464. doi: 10.1016/j.jde.2016.08.017. Google Scholar [17] J. J. Kohn, Hypoellipticity of some degenerate subelliptic operators, J. Funct. Anal., 159 (1998), 203-216. doi: 10.1006/jfan.1998.3289. Google Scholar [18] J. J. Kohn, Hypoellipticity at points of infinite type, Proceedings of the International Conference on Analysis, Geometry, Number Theory in honor of Leon Ehrenpreis (Philadelphia, 1998), Contemp. Math., 251 (2000), 393-398. doi: 10.1090/conm/251/03882. Google Scholar [19] M. Koike, A note on hypoellipticity for degenerate elliptic operators, Publ. Res. Inst. Math. Sci., 27 (1991), 995-1000. doi: 10.2977/prims/1195169008. Google Scholar [20] V. Komornik, Exact Controllability and Stabilization, The Multiplier Method, Mason-John Wiley, Paris, 1994. Google Scholar [21] M. O. Korpusov and A. G. Sveshnikov, Three-dimensional nonlinear evolution equations of pseudoparabolic type in problems of mathematical physics, Zh. Vychisl. Mat. Mat. Fiz., 43 (2003), 1835-1869 (in Russian); transl. in: Comput. Math. Math. Phys., 43 (2003), 1765-1797. Google Scholar [22] M. O. Korpusov and A. G. Sveshnikov, Blow-up of solutions of Sobolev-type nonlinear equations with cubic sources, Differ. Uravn., 42 (2006), 404-415 (in Russian); transl. in: Differ. Equ., 42 (2006), 431-443. doi: 10.1134/S001226610603013X. Google Scholar [23] 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 [24] Y. Liu and J. Zhao, On potential wells and applications to semilinear hyperbolic equations and parabolic equations, Nonlinear Anal., 64 (2006), 2665-2687. doi: 10.1016/j.na.2005.09.011. Google Scholar [25] Y. Morimoto, On the hypoellipticity for infinitely degenerate semi-elliptic operators, J. Math. Soc. Jpn., 30 (1978), 327-358. doi: 10.2969/jmsj/03020327. Google Scholar [26] Y. Morimoto, A criterion for hypoellipticity of second order differential operators, Osaka J. Math., 24 (1987), 651-675. Google Scholar [27] Y. Morimoto, Hypoellipticity for infinitely degenerate elliptic operators, Osaka J. Math., 24 (1987), 13-35. Google Scholar [28] Y. Morimoto and T. Morioka, The positivity of Schrödinger operators and the hypoellipticity of second order degenerate elliptic operators, Bull. Sci. Math., 121 (1997), 507-547. Google Scholar [29] Y. Morimoto and T. Morioka, Hypoellipticity for elliptic operators with infinite degeneracy, Partial Differential Equations and Their Applications (H. Chen and L. Rodino, eds.), World Sci. Publishing, River Edge, NJ, (1999), 240-259. Google Scholar [30] Y. Morimoto and C. Xu, Logarithmic Sobolev inequality and semi-linear Dirichlet problems for infinitely degenerate elliptic operators, Astérisque, 284 (2003), 245-264. Google Scholar [31] V. Padrón, Effect of aggregation on population recovery modeled by a forward-backward pseudoparabolic equation, Trans. Amer. Math. Soc., 356 (2004), 2739-2756. doi: 10.1090/S0002-9947-03-03340-3. Google Scholar [32] L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Isr. J. Math., 22 (1975), 273-303. doi: 10.1007/BF02761595. Google Scholar [33] M. Ptashnyk, Degenerate quasilinear pseudoparabolic equations with memory terms and variational inequalities, Nonlinear Anal., 66 (2007), 2653-2675. doi: 10.1016/j.na.2006.03.046. Google Scholar [34] D. H. Sattinger, On global solution of nonlinear hyperbolic equations, Arch. Ration. Mech. Anal., 30 (1968), 148-172. doi: 10.1007/BF00250942. Google Scholar [35] R. E. Showalter and T. W. Ting, Pseudoparabolic partial differential equations, SIAM J. Math. Anal., 1 (1970), 1-26. doi: 10.1137/0501001. Google Scholar [36] S. L. Sobolev, On a new problem of mathematical physics, Izv. Akad. Nauk SSSR Ser. Math., 18 (1954), 3-50. Google Scholar [37] T. W. Ting, Certain non-steady flows of second-order fluids, Arch. Ration. Mech. Anal., 14 (1963), 1-26. doi: 10.1007/BF00250690. Google Scholar [38] T. W. Ting, Parabolic and pseudo-parabolic partial differential equations, J. Math. Soc. Japan, 21 (1969), 440-453. doi: 10.2969/jmsj/02130440. Google Scholar [39] R. Xu and J. Su, Global existence and finite time blow-up for a class of semilinear pseudo-parabolic equations, J. Funct. Anal., 264 (2013), 2732-2763. doi: 10.1016/j.jfa.2013.03.010. Google Scholar

show all references

References:
 [1] J. M. Ball, Remarks on blow-up and nonexistence theorems for nonlinear evolution equations, Quart. J. Math. Oxford Ser., 28 (1977), 473-486. doi: 10.1093/qmath/28.4.473. Google Scholar [2] G. Barenblat, I. Zheltov and I. Kochiva, Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks, J. Appl. Math. Mech., 24 (1960), 1286-1303. Google Scholar [3] E. D. Benedetto and M. Pierre, On the maximum principle for pseudoparabolic equations, Indiana Univ. Math. J., 30 (1981), 821-854. doi: 10.1512/iumj.1981.30.30062. Google Scholar [4] T. B. Benjamin, J. L. Bona and J. J. Mahony, Model equations for long waves in nonlinear dispersive systems, Philos. Trans. R. Soc. Lond. Ser. A, 272 (1972), 47-78. doi: 10.1098/rsta.1972.0032. Google Scholar [5] H. Brill, A semilinear Sobolev evolution equation in a Banach space, J. Differential Equations, 24 (1977), 412-425. doi: 10.1016/0022-0396(77)90009-2. Google Scholar [6] Y. Cao, J. Yin and C. Wang, Cauchy problems of semilinear pseudo-parabolic equations, J. Differential Equations, 246 (2009), 4568-4590. doi: 10.1016/j.jde.2009.03.021. Google Scholar [7] H. Chen and K. Li, The existence and regularity of multiple solutions for a class of infinitely degenerate elliptic equations, Math Nach, 282 (2009), 368-385. doi: 10.1002/mana.200710742. Google Scholar [8] H. Chen, P. Luo and G. Liu, Global solution and blow up of a semilinear heat equation with logarithmic nonlinearity, J. Math. Anal. Appl., 422 (2015), 84-98. doi: 10.1016/j.jmaa.2014.08.030. Google Scholar [9] H. Chen, P. Luo and S. Tian, Existence and regularity of multiple solutions forinfinitely degenerate nonlinear elliptic equations with singular potential, J. Differ Equ., 257 (2014), 3300-3333. doi: 10.1016/j.jde.2014.06.014. Google Scholar [10] H. Chen, P. Luo and S. Tian, Multiplicity and regularity of solutions for infinitely degenerate elliptic equations with a free perturbation, J. Math. Pures Appl., 103 (2015), 849-867. doi: 10.1016/j.matpur.2014.09.004. Google Scholar [11] H. Chen and S. Tian, Initial boundary value problem for a class of semilinear pseudo-parabolic equations with logarithmic nonlinearity, J. Differential Equations, 285 (2015), 4424-4442. doi: 10.1016/j.jde.2015.01.038. Google Scholar [12] M. Christ, Hypoellipticity in the infinitely degenerate regime, Complex Analysis and Geometry (Columbus, OH, 1999), 59-84, Ohio State Univ. Math. Res. Inst. Publ., 9, de Gruyter, Berlin, 2001. Google Scholar [13] C. David and W. Jet, Asymptotic behaviour of the fundamental solution to the equation of heat conduction in two temperatures, J. Math. Anal. Appl., 69 (1979), 411-418. doi: 10.1016/0022-247X(79)90152-5. Google Scholar [14] V. R. Gopala Rao and T. W. Ting, Solutions of pseudo-heat equations in the whole space, Arch. Ration. Mech. Anal., 49 (1972/73), 57-78. doi: 10.1007/BF00281474. Google Scholar [15] L. Hörmander, Hypoelliptic second order differential equations, Acta Math., 119 (1967), 147-171. doi: 10.1007/BF02392081. Google Scholar [16] S. Ji, J. Yin and Y. Cao, Instability of positive periodic solutions for semilinear pseudo-parabolic equations with logarithmic nonlinearity, J. Differential Equations, 261 (2016), 5446-5464. doi: 10.1016/j.jde.2016.08.017. Google Scholar [17] J. J. Kohn, Hypoellipticity of some degenerate subelliptic operators, J. Funct. Anal., 159 (1998), 203-216. doi: 10.1006/jfan.1998.3289. Google Scholar [18] J. J. Kohn, Hypoellipticity at points of infinite type, Proceedings of the International Conference on Analysis, Geometry, Number Theory in honor of Leon Ehrenpreis (Philadelphia, 1998), Contemp. Math., 251 (2000), 393-398. doi: 10.1090/conm/251/03882. Google Scholar [19] M. Koike, A note on hypoellipticity for degenerate elliptic operators, Publ. Res. Inst. Math. Sci., 27 (1991), 995-1000. doi: 10.2977/prims/1195169008. Google Scholar [20] V. Komornik, Exact Controllability and Stabilization, The Multiplier Method, Mason-John Wiley, Paris, 1994. Google Scholar [21] M. O. Korpusov and A. G. Sveshnikov, Three-dimensional nonlinear evolution equations of pseudoparabolic type in problems of mathematical physics, Zh. Vychisl. Mat. Mat. Fiz., 43 (2003), 1835-1869 (in Russian); transl. in: Comput. Math. Math. Phys., 43 (2003), 1765-1797. Google Scholar [22] M. O. Korpusov and A. G. Sveshnikov, Blow-up of solutions of Sobolev-type nonlinear equations with cubic sources, Differ. Uravn., 42 (2006), 404-415 (in Russian); transl. in: Differ. Equ., 42 (2006), 431-443. doi: 10.1134/S001226610603013X. Google Scholar [23] 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 [24] Y. Liu and J. Zhao, On potential wells and applications to semilinear hyperbolic equations and parabolic equations, Nonlinear Anal., 64 (2006), 2665-2687. doi: 10.1016/j.na.2005.09.011. Google Scholar [25] Y. Morimoto, On the hypoellipticity for infinitely degenerate semi-elliptic operators, J. Math. Soc. Jpn., 30 (1978), 327-358. doi: 10.2969/jmsj/03020327. Google Scholar [26] Y. Morimoto, A criterion for hypoellipticity of second order differential operators, Osaka J. Math., 24 (1987), 651-675. Google Scholar [27] Y. Morimoto, Hypoellipticity for infinitely degenerate elliptic operators, Osaka J. Math., 24 (1987), 13-35. Google Scholar [28] Y. Morimoto and T. Morioka, The positivity of Schrödinger operators and the hypoellipticity of second order degenerate elliptic operators, Bull. Sci. Math., 121 (1997), 507-547. Google Scholar [29] Y. Morimoto and T. Morioka, Hypoellipticity for elliptic operators with infinite degeneracy, Partial Differential Equations and Their Applications (H. Chen and L. Rodino, eds.), World Sci. Publishing, River Edge, NJ, (1999), 240-259. Google Scholar [30] Y. Morimoto and C. Xu, Logarithmic Sobolev inequality and semi-linear Dirichlet problems for infinitely degenerate elliptic operators, Astérisque, 284 (2003), 245-264. Google Scholar [31] V. Padrón, Effect of aggregation on population recovery modeled by a forward-backward pseudoparabolic equation, Trans. Amer. Math. Soc., 356 (2004), 2739-2756. doi: 10.1090/S0002-9947-03-03340-3. Google Scholar [32] L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Isr. J. Math., 22 (1975), 273-303. doi: 10.1007/BF02761595. Google Scholar [33] M. Ptashnyk, Degenerate quasilinear pseudoparabolic equations with memory terms and variational inequalities, Nonlinear Anal., 66 (2007), 2653-2675. doi: 10.1016/j.na.2006.03.046. Google Scholar [34] D. H. Sattinger, On global solution of nonlinear hyperbolic equations, Arch. Ration. Mech. Anal., 30 (1968), 148-172. doi: 10.1007/BF00250942. Google Scholar [35] R. E. Showalter and T. W. Ting, Pseudoparabolic partial differential equations, SIAM J. Math. Anal., 1 (1970), 1-26. doi: 10.1137/0501001. Google Scholar [36] S. L. Sobolev, On a new problem of mathematical physics, Izv. Akad. Nauk SSSR Ser. Math., 18 (1954), 3-50. Google Scholar [37] T. W. Ting, Certain non-steady flows of second-order fluids, Arch. Ration. Mech. Anal., 14 (1963), 1-26. doi: 10.1007/BF00250690. Google Scholar [38] T. W. Ting, Parabolic and pseudo-parabolic partial differential equations, J. Math. Soc. Japan, 21 (1969), 440-453. doi: 10.2969/jmsj/02130440. Google Scholar [39] R. Xu and J. Su, Global existence and finite time blow-up for a class of semilinear pseudo-parabolic equations, J. Funct. Anal., 264 (2013), 2732-2763. doi: 10.1016/j.jfa.2013.03.010. Google Scholar
 [1] Xiaoli Zhu, Fuyi Li, Ting Rong. Global existence and blow up of solutions to a class of pseudo-parabolic equations with an exponential source. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2465-2485. doi: 10.3934/cpaa.2015.14.2465 [2] Monica Marras, Stella Vernier-Piro, Giuseppe Viglialoro. Blow-up phenomena for nonlinear pseudo-parabolic equations with gradient term. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2291-2300. doi: 10.3934/dcdsb.2017096 [3] Ronghua Jiang, Jun Zhou. Blow-up and global existence of solutions to a parabolic equation associated with the fraction p-Laplacian. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1205-1226. doi: 10.3934/cpaa.2019058 [4] Long Wei, Zhijun Qiao, Yang Wang, Shouming Zhou. Conserved quantities, global existence and blow-up for a generalized CH equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1733-1748. doi: 10.3934/dcds.2017072 [5] Yang Cao, Jingxue Yin. Small perturbation of a semilinear pseudo-parabolic equation. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 631-642. doi: 10.3934/dcds.2016.36.631 [6] Monica Marras, Stella Vernier Piro. On global existence and bounds for blow-up time in nonlinear parabolic problems with time dependent coefficients. Conference Publications, 2013, 2013 (special) : 535-544. doi: 10.3934/proc.2013.2013.535 [7] Mu-Ming Zhang, Tian-Yuan Xu, Jing-Xue Yin. Controllability properties of degenerate pseudo-parabolic boundary control problems. Mathematical Control & Related Fields, 2019, 0 (0) : 0-0. doi: 10.3934/mcrf.2019034 [8] Huafei Di, Yadong Shang, Xiaoxiao Zheng. Global well-posedness for a fourth order pseudo-parabolic equation with memory and source terms. Discrete & Continuous Dynamical Systems - B, 2016, 21 (3) : 781-801. doi: 10.3934/dcdsb.2016.21.781 [9] Shuyin Wu, Joachim Escher, Zhaoyang Yin. Global existence and blow-up phenomena for a weakly dissipative Degasperis-Procesi equation. Discrete & Continuous Dynamical Systems - B, 2009, 12 (3) : 633-645. doi: 10.3934/dcdsb.2009.12.633 [10] Jong-Shenq Guo. Blow-up behavior for a quasilinear parabolic equation with nonlinear boundary condition. Discrete & Continuous Dynamical Systems - A, 2007, 18 (1) : 71-84. doi: 10.3934/dcds.2007.18.71 [11] Sachiko Ishida, Tomomi Yokota. Blow-up in finite or infinite time for quasilinear degenerate Keller-Segel systems of parabolic-parabolic type. Discrete & Continuous Dynamical Systems - B, 2013, 18 (10) : 2569-2596. doi: 10.3934/dcdsb.2013.18.2569 [12] Shouming Zhou, Chunlai Mu, Liangchen Wang. Well-posedness, blow-up phenomena and global existence for the generalized $b$-equation with higher-order nonlinearities and weak dissipation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 843-867. doi: 10.3934/dcds.2014.34.843 [13] Guangyu Xu, Jun Zhou. Global existence and blow-up of solutions to a singular Non-Newton polytropic filtration equation with critical and supercritical initial energy. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1805-1820. doi: 10.3934/cpaa.2018086 [14] Hua Chen, Nian Liu. Asymptotic stability and blow-up of solutions for semi-linear edge-degenerate parabolic equations with singular potentials. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 661-682. doi: 10.3934/dcds.2016.36.661 [15] Victor A. Galaktionov, Juan-Luis Vázquez. The problem Of blow-up in nonlinear parabolic equations. Discrete & Continuous Dynamical Systems - A, 2002, 8 (2) : 399-433. doi: 10.3934/dcds.2002.8.399 [16] Shiming Li, Yongsheng Li, Wei Yan. A global existence and blow-up threshold for Davey-Stewartson equations in $\mathbb{R}^3$. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1899-1912. doi: 10.3934/dcdss.2016077 [17] Lili Du, Chunlai Mu, Zhaoyin Xiang. Global existence and blow-up to a reaction-diffusion system with nonlinear memory. Communications on Pure & Applied Analysis, 2005, 4 (4) : 721-733. doi: 10.3934/cpaa.2005.4.721 [18] Shu-Xiang Huang, Fu-Cai Li, Chun-Hong Xie. Global existence and blow-up of solutions to a nonlocal reaction-diffusion system. Discrete & Continuous Dynamical Systems - A, 2003, 9 (6) : 1519-1532. doi: 10.3934/dcds.2003.9.1519 [19] Zaihui Gan, Jian Zhang. Blow-up, global existence and standing waves for the magnetic nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2012, 32 (3) : 827-846. doi: 10.3934/dcds.2012.32.827 [20] Bin Li. On the blow-up criterion and global existence of a nonlinear PDE system in biological transport networks. Kinetic & Related Models, 2019, 12 (5) : 1131-1162. doi: 10.3934/krm.2019043

2018 Impact Factor: 1.143