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

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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 and Applied Analysis, 2020, 19 (2) : 923-939. doi: 10.3934/cpaa.2020042

References:

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[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[12]	L. Gross, Logarithmic Sobolev inequalities, Amer. J. Math., 97 (1975), 1061-1083. doi: 10.2307/2373688.
[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.
[14]	K. Hayakawa, On nonexistence of global solutions of some semilinear parabolic differential equations, Proc. Japan Acad., 49 (1973), 503-505.
[15]	D. Henry, Geometric Theory of Semilinear Parabolic Equations, volume 840 of Lecture Notes in Mathematics, Springer-Verlag, Berlin-New York, 1981.
[16]	H. Hoshino and Y. Yamada, Solvability and smoothing effect for semilinear parabolic equations, Funkcial. Ekvac., 34 (1991), 475-494.
[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.
[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.
[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.
[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.
[21]	J. L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod; Gauthier-Villars, Paris, 1969.
[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.
[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.
[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.
[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.
[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.
[27]	P. Quittner and P. Souplet, Superlinear Parabolic Problems, Blow-Up, Global Existence and Steady States, Birkhäuser Verlag, Basel, 2007.,
[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.
[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.
[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.
[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.
[32]	M. Tsutsumi, On solutions of semilinear differential equations in a Hilbert space, Math. Japon., 17 (1972), 173-193.
[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.
[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.
[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.
[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.
[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.
[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.

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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[12]	L. Gross, Logarithmic Sobolev inequalities, Amer. J. Math., 97 (1975), 1061-1083. doi: 10.2307/2373688.
[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.
[14]	K. Hayakawa, On nonexistence of global solutions of some semilinear parabolic differential equations, Proc. Japan Acad., 49 (1973), 503-505.
[15]	D. Henry, Geometric Theory of Semilinear Parabolic Equations, volume 840 of Lecture Notes in Mathematics, Springer-Verlag, Berlin-New York, 1981.
[16]	H. Hoshino and Y. Yamada, Solvability and smoothing effect for semilinear parabolic equations, Funkcial. Ekvac., 34 (1991), 475-494.
[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.
[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.
[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.
[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.
[21]	J. L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod; Gauthier-Villars, Paris, 1969.
[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.
[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.
[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.
[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.
[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.
[27]	P. Quittner and P. Souplet, Superlinear Parabolic Problems, Blow-Up, Global Existence and Steady States, Birkhäuser Verlag, Basel, 2007.,
[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.
[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.
[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.
[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.
[32]	M. Tsutsumi, On solutions of semilinear differential equations in a Hilbert space, Math. Japon., 17 (1972), 173-193.
[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.
[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.
[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.
[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.
[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.
[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.

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