American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Global solvability and general decay of a transmission problem for kirchhoff-type wave equations with nonlinear damping and delay term
  • CPAA Home
  • This Issue
  • Next Article
    Asymptotic profile of solutions to a certain chemotaxis system
February  2020, 19(2): 923-939. doi: 10.3934/cpaa.2020042

Global existence and blow-up of solutions to a semilinear heat equation with singular potential and logarithmic nonlinearity

Xiumei Deng and  Jun Zhou ,

School of Mathematics and Statistics, Southwest University, Chongqing, 400715, China

* Corresponding author: Jun Zhou

Received  February 2019 Revised  May 2019 Published  October 2019

Fund Project: The second author is supported by NSFC grant 11201380

This paper concerns with a semilinear heat equation with singular potential and logarithmic nonlinearity. By using the logarithmic Sobolev inequality and a family of potential wells, the existence of global solutions and infinite time blow-up solutions are obtained. The results of this paper indicate that the polynomial nonlinearity is a critical condition of existence of finite time blow-up solutions to semilinear heat equation with singular potential.

Keywords: Semilinear heat equation, singular potential, logarithmic nonlinearity, global existence, blow-up.
Mathematics Subject Classification: Primary: 35B40; Secondary: 35K58.
Citation: Xiumei Deng, Jun Zhou. Global existence and blow-up of solutions to a semilinear heat equation with singular potential and logarithmic nonlinearity. Communications on Pure & Applied Analysis, 2020, 19 (2) : 923-939. doi: 10.3934/cpaa.2020042
References:
[1]

A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis, 14 (1973), 349-381.  doi: 10.1016/0022-1236(73)90051-7.  Google Scholar

[2]

M. Badiale and G. Tarantello, A Sobolev-Hardy inequality with applications to a nonlinear elliptic equation arising in astrophysics, Arch. Ration. Mech. Anal., 163 (2002), 259-293.  doi: 10.1007/s002050200201.  Google Scholar

[3]

J. M. Ball, Remarks on blow-up and nonexistence theorems for nonlinear evolution equations, Quart. J. Math. Oxford Ser. (2), 28 (1977), 473–486. doi: 10.1093/qmath/28.4.473.  Google Scholar

[4]

K. Bouhali and F. Ellaggoune, Viscoelastic wave equation with logarithmic nonlinearities in $\Bbb R^n$, J. Partial Differ. Equ., 30 (2017), 47-63.  doi: 10.4208/jpde.v30.n1.4.  Google Scholar

[5]

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

[6]

H. Chen and S. Tian, Initial boundary value problem for a class of semilinear pseudo-parabolic equations with logarithmic nonlinearity, J. Differential Equations, 258 (2015), 4424-4442.  doi: 10.1016/j.jde.2015.01.038.  Google Scholar

[7]

L. C. Evans, Partial Differential Equations, second edition, volume 19 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/019.  Google Scholar

[8]

M. Feng and J. Zhou, Global existence and blow-up of solutions to a nonlocal parabolic equation with singular potential, J. Math. Anal. Appl., 464 (2018), 1213-1242.  doi: 10.1016/j.jmaa.2018.04.056.  Google Scholar

[9]

H. Fujita, On the blowing up of solutions of the Cauchy problem for $u_{t} = \Delta u+u^{1+\alpha }$, J. Fac. Sci. Univ. Tokyo Sect. I, 13 (1966), 109-124.   Google Scholar

[10]

F. Gazzola and T. Weth, Finite time blow-up and global solutions for semilinear parabolic equations with initial data at high energy level, Differential Integral Equations, 18 (2005), 961-990.   Google Scholar

[11]

Y. Giga, S. Matsui and S. Sasayama, Blow up rate for semilinear heat equations with subcritical nonlinearity, Indiana Univ. Math. J., 53 (2004), 483-514.  doi: 10.1512/iumj.2004.53.2401.  Google Scholar

[12]

L. Gross, Logarithmic Sobolev inequalities, Amer. J. Math., 97 (1975), 1061-1083.  doi: 10.2307/2373688.  Google Scholar

[13]

J. Hao and J. Zhou, A new blow-up condition for a parabolic equation with singular potential, J. Math. Anal. Appl., 449 (2017), 897-906.  doi: 10.1016/j.jmaa.2016.12.040.  Google Scholar

[14]

K. Hayakawa, On nonexistence of global solutions of some semilinear parabolic differential equations, Proc. Japan Acad., 49 (1973), 503-505.   Google Scholar

[15]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, volume 840 of Lecture Notes in Mathematics, Springer-Verlag, Berlin-New York, 1981.  Google Scholar

[16]

H. Hoshino and Y. Yamada, Solvability and smoothing effect for semilinear parabolic equations, Funkcial. Ekvac., 34 (1991), 475-494.   Google Scholar

[17]

C. Ji and A. Szulkin, A logarithmic Schrödinger equation with asymptotic conditions on the potential, J. Math. Anal. Appl., 437 (2016), 241-254.  doi: 10.1016/j.jmaa.2015.11.071.  Google Scholar

[18]

S. M. Ji, J. X. 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

[19]

H. A. Levine and J. Serrin, Global nonexistence theorems for quasilinear evolution equations with dissipation, Arch. Rational Mech. Anal., 137 (1997), 341-361.  doi: 10.1007/s002050050032.  Google Scholar

[20]

E. H. Lieb and M. Loss, Analysis, volume 14 of Graduate Studies in Mathematics, second edition, American Mathematical Society, Providence, RI, 2001. doi: 10.1090/gsm/014.  Google Scholar

[21]

J. L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod; Gauthier-Villars, Paris, 1969.  Google Scholar

[22]

H. L. Liu, Z. S. Liu and Q. Z. Xiao, Ground state solution for a fourth-order nonlinear elliptic problem with logarithmic nonlinearity, Appl. Math. Lett., 79 (2018), 176-181.  doi: 10.1016/j.aml.2017.12.015.  Google Scholar

[23]

Y. Q. Liu and W. K. Wang, Local well-posedness of a new integrable equation, Nonlinear Anal., 64 (2006), 2516-2526.  doi: 10.1016/j.na.2005.08.030.  Google Scholar

[24]

L. E. Payne, G. A. Philippin and P. W. Schaefer, Blow-up phenomena for some nonlinear parabolic problems, Nonlinear Anal., 69 (2008), 3495-3502.  doi: 10.1016/j.na.2007.09.035.  Google Scholar

[25]

L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Israel J. Math., 22 (1975), 273-303.  doi: 10.1007/BF02761595.  Google Scholar

[26]

L. E. Payne and P. W. Schaefer, Lower bounds for blow-up time in parabolic problems under Dirichlet conditions, J. Math. Anal. Appl., 328 (2007), 1196-1205.  doi: 10.1016/j.jmaa.2006.06.015.  Google Scholar

[27]

P. Quittner and P. Souplet, Superlinear Parabolic Problems, Blow-Up, Global Existence and Steady States, Birkhäuser Verlag, Basel, 2007.,  Google Scholar

[28]

M. Squassina and A. Szulkin, Multiple solutions to logarithmic Schrödinger equations with periodic potential, Calc. Var. Partial Differential Equations, 54 (2015), 585-597.  doi: 10.1007/s00526-014-0796-8.  Google Scholar

[29]

Z. Tan, Non-Newton filtration equation with special medium void, Acta Math. Sci. Ser. B (Engl. Ed.), 24 (2004), 118-128.  doi: 10.1016/S0252-9602(17)30367-3.  Google Scholar

[30]

K. Tanaka and C. X. Zhang, Multi-bump solutions for logarithmic Schrödinger equations, Calc. Var. Partial Differential Equations, 56 (2017), 33. doi: 10.1007/s00526-017-1122-z.  Google Scholar

[31]

S. Y. Tian, Multiple solutions for the semilinear elliptic equations with the sign-changing logarithmic nonlinearity, J. Math. Anal. Appl., 454 (2017), 816-828.  doi: 10.1016/j.jmaa.2017.05.015.  Google Scholar

[32]

M. Tsutsumi, On solutions of semilinear differential equations in a Hilbert space, Math. Japon., 17 (1972), 173-193.   Google Scholar

[33]

Y. Wang, The existence of global solution and the blowup problem for some $p$-Laplace heat equations, Acta Math. Sci. Ser. B (Engl. Ed.), 27 (2007), 274-282.  doi: 10.1016/S0252-9602(07)60026-5.  Google Scholar

[34]

F. B. Weissler, Semilinear evolution equations in Banach spaces, J. Funct. Anal., 32 (1979), 277-296.  doi: 10.1016/0022-1236(79)90040-5.  Google Scholar

[35]

F. B. Weissler, Existence and nonexistence of global solutions for a semilinear heat equation, Israel J. Math., 38 (1981), 29-40.  doi: 10.1007/BF02761845.  Google Scholar

[36]

G. Y. Xu and J. Zhou, Global existence and blow-up of solutions to a singular non-Newton polytropic filtration equation with critical and supercritical initial energy, Commun. Pure Appl. Anal., 17 (2018), 1805-1820.  doi: 10.3934/cpaa.2018086.  Google Scholar

[37]

J. Zhou, A multi-dimension blow-up problem to a porous medium diffusion equation with special medium void, Appl. Math. Lett., 30 (2014), 6-11.  doi: 10.1016/j.aml.2013.12.003.  Google Scholar

[38]

J. Zhou, Global existence and blow-up of solutions for a non-Newton polytropic filtration system with special volumetric moisture content, Comput. Math. Appl., 71 (2016), 1163-1172.  doi: 10.1016/j.camwa.2016.01.029.  Google Scholar

show all references

References:
[1]

A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis, 14 (1973), 349-381.  doi: 10.1016/0022-1236(73)90051-7.  Google Scholar

[2]

M. Badiale and G. Tarantello, A Sobolev-Hardy inequality with applications to a nonlinear elliptic equation arising in astrophysics, Arch. Ration. Mech. Anal., 163 (2002), 259-293.  doi: 10.1007/s002050200201.  Google Scholar

[3]

J. M. Ball, Remarks on blow-up and nonexistence theorems for nonlinear evolution equations, Quart. J. Math. Oxford Ser. (2), 28 (1977), 473–486. doi: 10.1093/qmath/28.4.473.  Google Scholar

[4]

K. Bouhali and F. Ellaggoune, Viscoelastic wave equation with logarithmic nonlinearities in $\Bbb R^n$, J. Partial Differ. Equ., 30 (2017), 47-63.  doi: 10.4208/jpde.v30.n1.4.  Google Scholar

[5]

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

[6]

H. Chen and S. Tian, Initial boundary value problem for a class of semilinear pseudo-parabolic equations with logarithmic nonlinearity, J. Differential Equations, 258 (2015), 4424-4442.  doi: 10.1016/j.jde.2015.01.038.  Google Scholar

[7]

L. C. Evans, Partial Differential Equations, second edition, volume 19 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/019.  Google Scholar

[8]

M. Feng and J. Zhou, Global existence and blow-up of solutions to a nonlocal parabolic equation with singular potential, J. Math. Anal. Appl., 464 (2018), 1213-1242.  doi: 10.1016/j.jmaa.2018.04.056.  Google Scholar

[9]

H. Fujita, On the blowing up of solutions of the Cauchy problem for $u_{t} = \Delta u+u^{1+\alpha }$, J. Fac. Sci. Univ. Tokyo Sect. I, 13 (1966), 109-124.   Google Scholar

[10]

F. Gazzola and T. Weth, Finite time blow-up and global solutions for semilinear parabolic equations with initial data at high energy level, Differential Integral Equations, 18 (2005), 961-990.   Google Scholar

[11]

Y. Giga, S. Matsui and S. Sasayama, Blow up rate for semilinear heat equations with subcritical nonlinearity, Indiana Univ. Math. J., 53 (2004), 483-514.  doi: 10.1512/iumj.2004.53.2401.  Google Scholar

[12]

L. Gross, Logarithmic Sobolev inequalities, Amer. J. Math., 97 (1975), 1061-1083.  doi: 10.2307/2373688.  Google Scholar

[13]

J. Hao and J. Zhou, A new blow-up condition for a parabolic equation with singular potential, J. Math. Anal. Appl., 449 (2017), 897-906.  doi: 10.1016/j.jmaa.2016.12.040.  Google Scholar

[14]

K. Hayakawa, On nonexistence of global solutions of some semilinear parabolic differential equations, Proc. Japan Acad., 49 (1973), 503-505.   Google Scholar

[15]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, volume 840 of Lecture Notes in Mathematics, Springer-Verlag, Berlin-New York, 1981.  Google Scholar

[16]

H. Hoshino and Y. Yamada, Solvability and smoothing effect for semilinear parabolic equations, Funkcial. Ekvac., 34 (1991), 475-494.   Google Scholar

[17]

C. Ji and A. Szulkin, A logarithmic Schrödinger equation with asymptotic conditions on the potential, J. Math. Anal. Appl., 437 (2016), 241-254.  doi: 10.1016/j.jmaa.2015.11.071.  Google Scholar

[18]

S. M. Ji, J. X. 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

[19]

H. A. Levine and J. Serrin, Global nonexistence theorems for quasilinear evolution equations with dissipation, Arch. Rational Mech. Anal., 137 (1997), 341-361.  doi: 10.1007/s002050050032.  Google Scholar

[20]

E. H. Lieb and M. Loss, Analysis, volume 14 of Graduate Studies in Mathematics, second edition, American Mathematical Society, Providence, RI, 2001. doi: 10.1090/gsm/014.  Google Scholar

[21]

J. L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod; Gauthier-Villars, Paris, 1969.  Google Scholar

[22]

H. L. Liu, Z. S. Liu and Q. Z. Xiao, Ground state solution for a fourth-order nonlinear elliptic problem with logarithmic nonlinearity, Appl. Math. Lett., 79 (2018), 176-181.  doi: 10.1016/j.aml.2017.12.015.  Google Scholar

[23]

Y. Q. Liu and W. K. Wang, Local well-posedness of a new integrable equation, Nonlinear Anal., 64 (2006), 2516-2526.  doi: 10.1016/j.na.2005.08.030.  Google Scholar

[24]

L. E. Payne, G. A. Philippin and P. W. Schaefer, Blow-up phenomena for some nonlinear parabolic problems, Nonlinear Anal., 69 (2008), 3495-3502.  doi: 10.1016/j.na.2007.09.035.  Google Scholar

[25]

L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Israel J. Math., 22 (1975), 273-303.  doi: 10.1007/BF02761595.  Google Scholar

[26]

L. E. Payne and P. W. Schaefer, Lower bounds for blow-up time in parabolic problems under Dirichlet conditions, J. Math. Anal. Appl., 328 (2007), 1196-1205.  doi: 10.1016/j.jmaa.2006.06.015.  Google Scholar

[27]

P. Quittner and P. Souplet, Superlinear Parabolic Problems, Blow-Up, Global Existence and Steady States, Birkhäuser Verlag, Basel, 2007.,  Google Scholar

[28]

M. Squassina and A. Szulkin, Multiple solutions to logarithmic Schrödinger equations with periodic potential, Calc. Var. Partial Differential Equations, 54 (2015), 585-597.  doi: 10.1007/s00526-014-0796-8.  Google Scholar

[29]

Z. Tan, Non-Newton filtration equation with special medium void, Acta Math. Sci. Ser. B (Engl. Ed.), 24 (2004), 118-128.  doi: 10.1016/S0252-9602(17)30367-3.  Google Scholar

[30]

K. Tanaka and C. X. Zhang, Multi-bump solutions for logarithmic Schrödinger equations, Calc. Var. Partial Differential Equations, 56 (2017), 33. doi: 10.1007/s00526-017-1122-z.  Google Scholar

[31]

S. Y. Tian, Multiple solutions for the semilinear elliptic equations with the sign-changing logarithmic nonlinearity, J. Math. Anal. Appl., 454 (2017), 816-828.  doi: 10.1016/j.jmaa.2017.05.015.  Google Scholar

[32]

M. Tsutsumi, On solutions of semilinear differential equations in a Hilbert space, Math. Japon., 17 (1972), 173-193.   Google Scholar

[33]

Y. Wang, The existence of global solution and the blowup problem for some $p$-Laplace heat equations, Acta Math. Sci. Ser. B (Engl. Ed.), 27 (2007), 274-282.  doi: 10.1016/S0252-9602(07)60026-5.  Google Scholar

[34]

F. B. Weissler, Semilinear evolution equations in Banach spaces, J. Funct. Anal., 32 (1979), 277-296.  doi: 10.1016/0022-1236(79)90040-5.  Google Scholar

[35]

F. B. Weissler, Existence and nonexistence of global solutions for a semilinear heat equation, Israel J. Math., 38 (1981), 29-40.  doi: 10.1007/BF02761845.  Google Scholar

[36]

G. Y. Xu and J. Zhou, Global existence and blow-up of solutions to a singular non-Newton polytropic filtration equation with critical and supercritical initial energy, Commun. Pure Appl. Anal., 17 (2018), 1805-1820.  doi: 10.3934/cpaa.2018086.  Google Scholar

[37]

J. Zhou, A multi-dimension blow-up problem to a porous medium diffusion equation with special medium void, Appl. Math. Lett., 30 (2014), 6-11.  doi: 10.1016/j.aml.2013.12.003.  Google Scholar

[38]

J. Zhou, Global existence and blow-up of solutions for a non-Newton polytropic filtration system with special volumetric moisture content, Comput. Math. Appl., 71 (2016), 1163-1172.  doi: 10.1016/j.camwa.2016.01.029.  Google Scholar

[1]

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
[2]

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
[3]

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
[4]

Alexander Gladkov. Blow-up problem for semilinear heat equation with nonlinear nonlocal Neumann boundary condition. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2053-2068. doi: 10.3934/cpaa.2017101
[5]

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
[6]

Van Tien Nguyen. On the blow-up results for a class of strongly perturbed semilinear heat equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3585-3626. doi: 10.3934/dcds.2015.35.3585
[7]

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
[8]

Yohei Fujishima. Blow-up set for a superlinear heat equation and pointedness of the initial data. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4617-4645. doi: 10.3934/dcds.2014.34.4617
[9]

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
[10]

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
[11]

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
[12]

Jianbo Cui, Jialin Hong, Liying Sun. On global existence and blow-up for damped stochastic nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6837-6854. doi: 10.3934/dcdsb.2019169
[13]

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
[14]

Keisuke Matsuya, Tetsuji Tokihiro. Existence and non-existence of global solutions for a discrete semilinear heat equation. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 209-220. doi: 10.3934/dcds.2011.31.209
[15]

Tarek Saanouni. A note on global well-posedness and blow-up of some semilinear evolution equations. Evolution Equations & Control Theory, 2015, 4 (3) : 355-372. doi: 10.3934/eect.2015.4.355
[16]

Jian Zhang, Shihui Zhu, Xiaoguang Li. Rate of $L^2$-concentration of the blow-up solution for critical nonlinear Schrödinger equation with potential. Mathematical Control & Related Fields, 2011, 1 (1) : 119-127. doi: 10.3934/mcrf.2011.1.119
[17]

Luigi Forcella, Kazumasa Fujiwara, Vladimir Georgiev, Tohru Ozawa. Local well-posedness and blow-up for the half Ginzburg-Landau-Kuramoto equation with rough coefficients and potential. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2661-2678. doi: 10.3934/dcds.2019111
[18]

Shota Sato. Blow-up at space infinity of a solution with a moving singularity for a semilinear parabolic equation. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1225-1237. doi: 10.3934/cpaa.2011.10.1225
[19]

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 - A, 2014, 34 (9) : 3831-3846. doi: 10.3934/dcds.2014.34.3831
[20]

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

2018 Impact Factor: 0.925

Tools

Metrics

  • PDF downloads (33)
  • HTML views (31)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences