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

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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

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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

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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

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

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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

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

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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

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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

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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

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

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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

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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

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[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

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