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Fractional Laplacians : A short survey
Existence and regularity results for a singular parabolic equations with degenerate coercivity
Laboratory LIPIM, National School of Applied Sciences Khouribga, Sultan Moulay Slimane University, Morocco |
$ \left\{ \begin{array}{lll} \dfrac{\partial u}{\partial t}-\mbox{div}\left( \dfrac{a(x,t,u,\nabla u)}{(1+|u|)^{\theta(p-1)}}\right) +g(x,t,u) = \dfrac{f}{u^{\gamma}} &\mbox{in}&\,\, Q,\\ u(x,0) = 0 &\mbox{on} & \Omega,\\ u = 0 &\mbox{on} &\,\, \Gamma. \end{array} \right. $ |
$ \Omega $ |
$ I\!\!R^{N} (N>p\geq 2), T>0 $ |
$ f $ |
$ f\in L^{m}(Q) $ |
$ Q = \Omega \times(0,T) $ |
$ \Gamma = \partial\Omega\times(0,T) $ |
$ g(x,t,u) = |u|^{s-1}u $ |
$ s\geq 1, $ |
$ 0\leq\theta< 1 $ |
$ 0<\gamma<1. $ |
References:
[1] |
L. Boccardo, A. Dall'Aglio, T. Gallouët and L. Orsina, Existence and regularity results for some nonlinear parabolic equations., Adv. Math. Sci. Appl., 9 (1999), 1017–1031. |
[2] |
L. Boccardo, A. Dall'Aglio, T. Gallouët and L. Orsina, Nonlinear parabolic equations with measure data, J. Funct. Anal., 147 (1997), 237–258.
doi: 10.1006/jfan.1996.3040. |
[3] |
L. Boccardo, T. Gallët and J. L. Vazquez, Solutions of nonlinear parabolic equations without growth restrictions on the data, Electron. J. Differential Equations, (2001), No.60, 20 pp. |
[4] |
G. R. Cirmi and M. M. Porzio, $L^{\infty}-$Solution for some nonlinear degenerate elliptic and parabolic equations, Ann. Mat. Pura Appl. (4), 169 (1995), 67–86.
doi: 10.1007/BF01759349. |
[5] |
A. Dall'Aglio and L. Orsina, Existence results for some nonlinear parabolic equations with nonregular data, Differential Integral Equations, 5 (1992), 1335–1354. |
[6] |
I. de Bonis and L. M. De Cave, Degenerate parabolic equations with singular lower order terms, Differential Integral Equations, 27 (2014), 949–976. |
[7] |
L. M. De Cave, Nonlinear elliptic equations with singular nonlinearities, Nonlinear Anal., 84 (2013), 181–195.
doi: 10.3233/ASY-131173. |
[8] |
L. M. De Cave and F. Oliva, On the regularizing effect of some absorption and sigular lower order trems in classical direchlet problems with $L^{1}$ data, J. Elliptic Parabol. Equ., 2 (2016), 73–85.
doi: 10.1007/BF03377393. |
[9] |
L. M. De Cave and F. Oliva, Elliptic equations with general singular lower order term and measure data, Nonlinear Anal., 128 (2015), 391–411.
doi: 10.1016/j.na.2015.08.005. |
[10] |
Y. El Hadfi, A. Benkirane and A. Youssfi, Existence and regularity results for parabolic equations with degenerate coercivity, Complex Var. Elliptic Equ., 63 (2018), 715–729.
doi: 10.1080/17476933.2017.1332596. |
[11] |
W. Fulks and J. S. Maybee, A singular non-linear equation, Osaka Math. J. 12 (1960), 1–19. |
[12] |
J. A. Gatica, V. Oliker and P. Waltman, Singular nonlinear boundary-value problems for second-order ordinary differential equations, J. Differential Equations, 79 (1989), 62–78.
doi: 10.1016/0022-0396(89)90113-7. |
[13] |
N. Grenon and A. Mercaldo,
Existence and regularity results for solutions to nonlinear parabolic equations, Adv. Differential Equations, 10 (2005), 1007-1034.
|
[14] |
F. Li, Existence and regularity results for some parabolic equations with degenerate coercivity, Ann. Acad. Sci. Fenn. Math., 37 (2012), 605–633.
doi: 10.5186/aasfm.2012.3738. |
[15] |
F.-Q. Li, Regularity of solutions to nonlinear parabolic equations with a lower-order term, Potential Anal. 16 (2002), 393–400.
doi: 10.1023/A:1014856614825. |
[16] |
J.-L. Lions, Quelques méthodes de résolutions des problÈmes aux limites nonlinéaires, Dunod, Gautthier-Villars, Paris, 1969. |
[17] |
A. Nachman and A. Challegari, A nonlinear singular boundary value problem in the theory of pseudoplastic fluids, SIAM J. Appl. Math., 38 (1980), 275–281.
doi: 10.1137/0138024. |
[18] |
L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 13 (1959), 115–162. |
[19] |
P. Nowsad, On the integral equation $\kappa f = 1/f$ arising in a problem in communication, J. Math. Anal. Appl., 14 (1966), 484–492.
doi: 10.1016/0022-247X(66)90008-4. |
[20] |
F. Oliva and F. Petitta, A nonlinear parabolic problem with singular terms and nonregular data, Nonlinear Anal., 194 (2020), 111472, 13 pp.
doi: 10.1016/j.na.2019.02.025. |
[21] |
F. Oliva and F. Pettita, On singular elliptic equations with measures sources, ESAIM Control Optim. Calc. Var., 22 (2016), 289–308.
doi: 10.1051/cocv/2015004. |
[22] |
A. Sbai and Y. El hadfi, Degenerate elliptic problem with a singular nonlinearity, arXiv: 2005.08383. |
[23] |
A. Sbai and Y. El hadfi, Regularizing effect of absorption terms in singular and degenerate elliptic problems, arXiv: 2008.03597. |
[24] |
J. Simon, Compact sets in the space $L^{p}(0, T; B).$, Ann. Mat. Pura Appl., 146 (1987), 65–96.
doi: 10.1007/BF01762360. |
[25] |
G. Stampacchia, Le probléme de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus, Ann. Inst. Fourier (Grenoble), 15 (1965), 189–258.
doi: 10.5802/aif.204. |
[26] |
A. Youssfi, A. Benkirane and Y. EL Hadfi, On bounded solutions for nonlinear parabolic equations with degenerate coercivity, Mediterr. J. Math., 13 (2016), 3029–3040.
doi: 10.1007/s00009-015-0670-8. |
show all references
References:
[1] |
L. Boccardo, A. Dall'Aglio, T. Gallouët and L. Orsina, Existence and regularity results for some nonlinear parabolic equations., Adv. Math. Sci. Appl., 9 (1999), 1017–1031. |
[2] |
L. Boccardo, A. Dall'Aglio, T. Gallouët and L. Orsina, Nonlinear parabolic equations with measure data, J. Funct. Anal., 147 (1997), 237–258.
doi: 10.1006/jfan.1996.3040. |
[3] |
L. Boccardo, T. Gallët and J. L. Vazquez, Solutions of nonlinear parabolic equations without growth restrictions on the data, Electron. J. Differential Equations, (2001), No.60, 20 pp. |
[4] |
G. R. Cirmi and M. M. Porzio, $L^{\infty}-$Solution for some nonlinear degenerate elliptic and parabolic equations, Ann. Mat. Pura Appl. (4), 169 (1995), 67–86.
doi: 10.1007/BF01759349. |
[5] |
A. Dall'Aglio and L. Orsina, Existence results for some nonlinear parabolic equations with nonregular data, Differential Integral Equations, 5 (1992), 1335–1354. |
[6] |
I. de Bonis and L. M. De Cave, Degenerate parabolic equations with singular lower order terms, Differential Integral Equations, 27 (2014), 949–976. |
[7] |
L. M. De Cave, Nonlinear elliptic equations with singular nonlinearities, Nonlinear Anal., 84 (2013), 181–195.
doi: 10.3233/ASY-131173. |
[8] |
L. M. De Cave and F. Oliva, On the regularizing effect of some absorption and sigular lower order trems in classical direchlet problems with $L^{1}$ data, J. Elliptic Parabol. Equ., 2 (2016), 73–85.
doi: 10.1007/BF03377393. |
[9] |
L. M. De Cave and F. Oliva, Elliptic equations with general singular lower order term and measure data, Nonlinear Anal., 128 (2015), 391–411.
doi: 10.1016/j.na.2015.08.005. |
[10] |
Y. El Hadfi, A. Benkirane and A. Youssfi, Existence and regularity results for parabolic equations with degenerate coercivity, Complex Var. Elliptic Equ., 63 (2018), 715–729.
doi: 10.1080/17476933.2017.1332596. |
[11] |
W. Fulks and J. S. Maybee, A singular non-linear equation, Osaka Math. J. 12 (1960), 1–19. |
[12] |
J. A. Gatica, V. Oliker and P. Waltman, Singular nonlinear boundary-value problems for second-order ordinary differential equations, J. Differential Equations, 79 (1989), 62–78.
doi: 10.1016/0022-0396(89)90113-7. |
[13] |
N. Grenon and A. Mercaldo,
Existence and regularity results for solutions to nonlinear parabolic equations, Adv. Differential Equations, 10 (2005), 1007-1034.
|
[14] |
F. Li, Existence and regularity results for some parabolic equations with degenerate coercivity, Ann. Acad. Sci. Fenn. Math., 37 (2012), 605–633.
doi: 10.5186/aasfm.2012.3738. |
[15] |
F.-Q. Li, Regularity of solutions to nonlinear parabolic equations with a lower-order term, Potential Anal. 16 (2002), 393–400.
doi: 10.1023/A:1014856614825. |
[16] |
J.-L. Lions, Quelques méthodes de résolutions des problÈmes aux limites nonlinéaires, Dunod, Gautthier-Villars, Paris, 1969. |
[17] |
A. Nachman and A. Challegari, A nonlinear singular boundary value problem in the theory of pseudoplastic fluids, SIAM J. Appl. Math., 38 (1980), 275–281.
doi: 10.1137/0138024. |
[18] |
L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 13 (1959), 115–162. |
[19] |
P. Nowsad, On the integral equation $\kappa f = 1/f$ arising in a problem in communication, J. Math. Anal. Appl., 14 (1966), 484–492.
doi: 10.1016/0022-247X(66)90008-4. |
[20] |
F. Oliva and F. Petitta, A nonlinear parabolic problem with singular terms and nonregular data, Nonlinear Anal., 194 (2020), 111472, 13 pp.
doi: 10.1016/j.na.2019.02.025. |
[21] |
F. Oliva and F. Pettita, On singular elliptic equations with measures sources, ESAIM Control Optim. Calc. Var., 22 (2016), 289–308.
doi: 10.1051/cocv/2015004. |
[22] |
A. Sbai and Y. El hadfi, Degenerate elliptic problem with a singular nonlinearity, arXiv: 2005.08383. |
[23] |
A. Sbai and Y. El hadfi, Regularizing effect of absorption terms in singular and degenerate elliptic problems, arXiv: 2008.03597. |
[24] |
J. Simon, Compact sets in the space $L^{p}(0, T; B).$, Ann. Mat. Pura Appl., 146 (1987), 65–96.
doi: 10.1007/BF01762360. |
[25] |
G. Stampacchia, Le probléme de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus, Ann. Inst. Fourier (Grenoble), 15 (1965), 189–258.
doi: 10.5802/aif.204. |
[26] |
A. Youssfi, A. Benkirane and Y. EL Hadfi, On bounded solutions for nonlinear parabolic equations with degenerate coercivity, Mediterr. J. Math., 13 (2016), 3029–3040.
doi: 10.1007/s00009-015-0670-8. |
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