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

Xiumei Deng; Jun Zhou

doi:10.3934/cpaa.2020042

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, 2019, 39 (2) : 1185-1203. doi: 10.3934/dcds.2019051
[2]	Asato Mukai, Yukihiro Seki. Refined construction of type II blow-up solutions for semilinear heat equations with Joseph–Lundgren supercritical nonlinearity. Discrete & Continuous Dynamical Systems, 2021 doi: 10.3934/dcds.2021060
[3]	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
[4]	Gongwei Liu. The existence, general decay and blow-up for a plate equation with nonlinear damping and a logarithmic source term. Electronic Research Archive, 2020, 28 (1) : 263-289. doi: 10.3934/era.2020016
[5]	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
[6]	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
[7]	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
[8]	Van Tien Nguyen. On the blow-up results for a class of strongly perturbed semilinear heat equations. Discrete & Continuous Dynamical Systems, 2015, 35 (8) : 3585-3626. doi: 10.3934/dcds.2015.35.3585
[9]	Mingyou Zhang, Qingsong Zhao, Yu Liu, Wenke Li. Finite time blow-up and global existence of solutions for semilinear parabolic equations with nonlinear dynamical boundary condition. Electronic Research Archive, 2020, 28 (1) : 369-381. doi: 10.3934/era.2020021
[10]	Xiaoliang Li, Baiyu Liu. Finite time blow-up and global solutions for a nonlocal parabolic equation with Hartree type nonlinearity. Communications on Pure & Applied Analysis, 2020, 19 (6) : 3093-3112. doi: 10.3934/cpaa.2020134
[11]	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
[12]	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
[13]	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
[14]	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
[15]	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
[16]	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
[17]	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
[18]	Justin Holmer, Chang Liu. Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity II: Supercritical blow-up profiles. Communications on Pure & Applied Analysis, 2021, 20 (1) : 215-242. doi: 10.3934/cpaa.2020264
[19]	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, 2014, 34 (2) : 843-867. doi: 10.3934/dcds.2014.34.843
[20]	Keisuke Matsuya, Tetsuji Tokihiro. Existence and non-existence of global solutions for a discrete semilinear heat equation. Discrete & Continuous Dynamical Systems, 2011, 31 (1) : 209-220. doi: 10.3934/dcds.2011.31.209

American Institute of Mathematical Sciences

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

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

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

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors