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July  2020, 19(7): 3625-3650. doi: 10.3934/cpaa.2020160

The Cauchy problem for heat equation with fractional Laplacian and exponential nonlinearity

Ahmad Z. Fino 1,, and  Mokhtar Kirane 2,

1. 

LaMA-Liban, Lebanese University, Faculty of Sciences, Department of Mathematics, P.O. Box 37 Tripoli, Lebanon
2. 

LaSIE, Pôle Sciences et Technologies, Université de La Rochelle, Avenue Michel Crépeau, 17031 La Rochelle, France

*Corresponding author

Received  July 2019 Revised  January 2020 Published  April 2020

We consider the Cauchy problem for heat equation with fractional Laplacian and exponential nonlinearity. We establish local well-posedness result in Orlicz spaces. We derive the existence of global solutions for small initial data. We obtain decay estimates for large time in Lebesgue spaces.

Keywords: Orlicz spaces, fractional Laplacian, well-posedness, global existence, decay estimates.
Mathematics Subject Classification: Primary: 35K05, 46E30, 35A01; Secondary: 35B40, 26A33, 35K55.
Citation: Ahmad Z. Fino, Mokhtar Kirane. The Cauchy problem for heat equation with fractional Laplacian and exponential nonlinearity. Communications on Pure and Applied Analysis, 2020, 19 (7) : 3625-3650. doi: 10.3934/cpaa.2020160
References:
[1] R. A. Adams and J. J. F. Fournier, Sobolev Spaces, 2$^nd$ edition, Pure and Applied Mathematics (Amsterdam), Vol. 140, Elsevier/Academic Press, Amsterdam, 2003. 
[2] C. Bennett and R. Sharpley, Interpolation of Operators, Pure and applied mathematics, Academic Press, 1988. 
[3]

Z. W. Birnbaum and W. Orlicz, \"Uber die Verallgemeinerung des Begriffes der zueinander konjugierten Potenzen, Studia Math., 3 (1931), 1-67. 

[4]

H. Brezis and T. Cazenave, A nonlinear heat equation with singular initial data, J. Anal. Math., 68 (1996), 277-304.  doi: 10.1007/BF02790212.

[5]

T. Cazenave and A. Haraux, Introduction aux Problémes d'évolution Semi-linéaires, Ellipses, Paris, 1990.

[6]

G. Furioli, T. Kawakami, B. Ruf and E. Terraneo, Asymptotic behavior and decay estimates of the solutions for a nonlinear parabolic equation with exponential nonlinearity, J. Differ. Equ., 262 (2017), 145-180.  doi: 10.1016/j.jde.2016.09.024.

[7]

S. Ibrahim, M. Majdoub and N. Masmoudi, Global solutions for a semilinear, two-dimensional Klein-Gordon equation with exponential-type, Commun. Pure Appl. Math., 59 (2006), 1639-1658.  doi: 10.1002/cpa.20127.

[8]

N. Ioku, The Cauchy problem for heat equations with exponential nonlinearity, J. Differ. Equ., 251 (2011), 1172-1194.  doi: 10.1016/j.jde.2011.02.015.

[9]

N. Ioku, B. Ruf and E. Terraneo, Existence, non-existence, and uniqueness for a heat equation with exponential nonlinearity in $\mathbb{R}^2$, Math. Phys. Anal. Geom., 18 (2015), Art. 29, 19 pp. doi: 10.1007/s11040-015-9199-0.

[10]

M. Majdoub, S. Otsmane and S. Tayachi, Local well-posedness and global existence for the biharmonic heat equation with exponential nonlinearity, Adv. Differ. Equ., 23 (2018), 489-522. 

[11]

M. Majdoub and S. Tayachi, Well-posedness, global existence and decay estimates for the heat equation with general power-exponential nonlinearities, Proc. Int. Cong. Math. Rio de Janeiro, 2 (2018), 2379-2404. 

[12]

M. Majdoub and S. Tayachi, Global existence and decay estimates for the heat equation with exponential nonlinearity, preprint, arXiv: 1912.06490v1.

[13]

M. M. Rao and Z. D. Ren, Applications of Orlicz Spaces, Monographs and Textbooks in Pure and Applied Mathematics, Vol. 250, Marcel Dekker, Inc., New York, 2002. doi: 10.1201/9780203910863.

[14]

N. S. Trudinger, On imbeddings into Orlicz spaces and some applications, J. Math. Mech., 17 (1967), 473-483.  doi: 10.1512/iumj.1968.17.17028.

[15]

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

[16]

F. B. Weissler, Local existence and nonexistence for semilinear parabolic equations in $L^p$, J. Indiana Univ. Math., 29 (1980), 79-102.  doi: 10.1512/iumj.1980.29.29007.

[17]

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.

show all references

References:
[1] R. A. Adams and J. J. F. Fournier, Sobolev Spaces, 2$^nd$ edition, Pure and Applied Mathematics (Amsterdam), Vol. 140, Elsevier/Academic Press, Amsterdam, 2003. 
[2] C. Bennett and R. Sharpley, Interpolation of Operators, Pure and applied mathematics, Academic Press, 1988. 
[3]

Z. W. Birnbaum and W. Orlicz, \"Uber die Verallgemeinerung des Begriffes der zueinander konjugierten Potenzen, Studia Math., 3 (1931), 1-67. 

[4]

H. Brezis and T. Cazenave, A nonlinear heat equation with singular initial data, J. Anal. Math., 68 (1996), 277-304.  doi: 10.1007/BF02790212.

[5]

T. Cazenave and A. Haraux, Introduction aux Problémes d'évolution Semi-linéaires, Ellipses, Paris, 1990.

[6]

G. Furioli, T. Kawakami, B. Ruf and E. Terraneo, Asymptotic behavior and decay estimates of the solutions for a nonlinear parabolic equation with exponential nonlinearity, J. Differ. Equ., 262 (2017), 145-180.  doi: 10.1016/j.jde.2016.09.024.

[7]

S. Ibrahim, M. Majdoub and N. Masmoudi, Global solutions for a semilinear, two-dimensional Klein-Gordon equation with exponential-type, Commun. Pure Appl. Math., 59 (2006), 1639-1658.  doi: 10.1002/cpa.20127.

[8]

N. Ioku, The Cauchy problem for heat equations with exponential nonlinearity, J. Differ. Equ., 251 (2011), 1172-1194.  doi: 10.1016/j.jde.2011.02.015.

[9]

N. Ioku, B. Ruf and E. Terraneo, Existence, non-existence, and uniqueness for a heat equation with exponential nonlinearity in $\mathbb{R}^2$, Math. Phys. Anal. Geom., 18 (2015), Art. 29, 19 pp. doi: 10.1007/s11040-015-9199-0.

[10]

M. Majdoub, S. Otsmane and S. Tayachi, Local well-posedness and global existence for the biharmonic heat equation with exponential nonlinearity, Adv. Differ. Equ., 23 (2018), 489-522. 

[11]

M. Majdoub and S. Tayachi, Well-posedness, global existence and decay estimates for the heat equation with general power-exponential nonlinearities, Proc. Int. Cong. Math. Rio de Janeiro, 2 (2018), 2379-2404. 

[12]

M. Majdoub and S. Tayachi, Global existence and decay estimates for the heat equation with exponential nonlinearity, preprint, arXiv: 1912.06490v1.

[13]

M. M. Rao and Z. D. Ren, Applications of Orlicz Spaces, Monographs and Textbooks in Pure and Applied Mathematics, Vol. 250, Marcel Dekker, Inc., New York, 2002. doi: 10.1201/9780203910863.

[14]

N. S. Trudinger, On imbeddings into Orlicz spaces and some applications, J. Math. Mech., 17 (1967), 473-483.  doi: 10.1512/iumj.1968.17.17028.

[15]

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

[16]

F. B. Weissler, Local existence and nonexistence for semilinear parabolic equations in $L^p$, J. Indiana Univ. Math., 29 (1980), 79-102.  doi: 10.1512/iumj.1980.29.29007.

[17]

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.

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