January  2022, 15(1): 117-141. doi: 10.3934/dcdss.2021012

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

* Corresponding author: yelhadfi@gmail.com

Received  August 2020 Revised  January 2021 Published  January 2022 Early access  January 2021

The aim of this paper is to prove existence and regularity of solutions for the following nonlinear singular parabolic problem
$ \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. $
Here
$ \Omega $
is a bounded open subset of
$ I\!\!R^{N} (N>p\geq 2), T>0 $
and
$ f $
is a non-negative function that belong to some Lebesgue space,
$ 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 $
and
$ 0<\gamma<1. $
Citation: Mounim El Ouardy, Youssef El Hadfi, Aziz Ifzarne. Existence and regularity results for a singular parabolic equations with degenerate coercivity. Discrete and Continuous Dynamical Systems - S, 2022, 15 (1) : 117-141. doi: 10.3934/dcdss.2021012
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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