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A lower bound for the principal eigenvalue of fully nonlinear elliptic operators
The Cauchy problem for heat equation with fractional Laplacian and exponential nonlinearity
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 |
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.
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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