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Global existence and blow-up of solutions to a semilinear heat equation with singular potential and logarithmic nonlinearity
School of Mathematics and Statistics, Southwest University, Chongqing, 400715, China |
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.
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. |
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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