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doi: 10.3934/dcdss.2021012

Existence and regularity results for a singular parabolic equations with degenerate coercivity

Mounim El Ouardy , Youssef El Hadfi , and  Aziz Ifzarne

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 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. $
Keywords: Singular parabolic problems, degenerate coercivity, existence, regularity, lower order term.
Mathematics Subject Classification: Primary: 35A21, 35B25, 35B45, 35B65, 35D30, 35K55, 35K65, 35K67; Secondary: 35Q35.
Citation: Mounim El Ouardy, Youssef El Hadfi, Aziz Ifzarne. Existence and regularity results for a singular parabolic equations with degenerate coercivity. Discrete & Continuous Dynamical Systems - S, 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

A. Dall'Aglio and L. Orsina, Existence results for some nonlinear parabolic equations with nonregular data, Differential Integral Equations, 5 (1992), 1335–1354.  Google Scholar

[6]

I. de Bonis and L. M. De Cave, Degenerate parabolic equations with singular lower order terms, Differential Integral Equations, 27 (2014), 949–976.  Google Scholar

[7]

L. M. De Cave, Nonlinear elliptic equations with singular nonlinearities, Nonlinear Anal., 84 (2013), 181–195. doi: 10.3233/ASY-131173.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

W. Fulks and J. S. Maybee, A singular non-linear equation, Osaka Math. J. 12 (1960), 1–19.  Google Scholar

[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.  Google Scholar

[13]

N. Grenon and A. Mercaldo, Existence and regularity results for solutions to nonlinear parabolic equations, Adv. Differential Equations, 10 (2005), 1007-1034.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

J.-L. Lions, Quelques méthodes de résolutions des problÈmes aux limites nonlinéaires, Dunod, Gautthier-Villars, Paris, 1969.  Google Scholar

[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.  Google Scholar

[18]

L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 13 (1959), 115–162.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

A. Sbai and Y. El hadfi, Degenerate elliptic problem with a singular nonlinearity, arXiv: 2005.08383. Google Scholar

[23]

A. Sbai and Y. El hadfi, Regularizing effect of absorption terms in singular and degenerate elliptic problems, arXiv: 2008.03597. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

A. Dall'Aglio and L. Orsina, Existence results for some nonlinear parabolic equations with nonregular data, Differential Integral Equations, 5 (1992), 1335–1354.  Google Scholar

[6]

I. de Bonis and L. M. De Cave, Degenerate parabolic equations with singular lower order terms, Differential Integral Equations, 27 (2014), 949–976.  Google Scholar

[7]

L. M. De Cave, Nonlinear elliptic equations with singular nonlinearities, Nonlinear Anal., 84 (2013), 181–195. doi: 10.3233/ASY-131173.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

W. Fulks and J. S. Maybee, A singular non-linear equation, Osaka Math. J. 12 (1960), 1–19.  Google Scholar

[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.  Google Scholar

[13]

N. Grenon and A. Mercaldo, Existence and regularity results for solutions to nonlinear parabolic equations, Adv. Differential Equations, 10 (2005), 1007-1034.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

J.-L. Lions, Quelques méthodes de résolutions des problÈmes aux limites nonlinéaires, Dunod, Gautthier-Villars, Paris, 1969.  Google Scholar

[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.  Google Scholar

[18]

L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 13 (1959), 115–162.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

A. Sbai and Y. El hadfi, Degenerate elliptic problem with a singular nonlinearity, arXiv: 2005.08383. Google Scholar

[23]

A. Sbai and Y. El hadfi, Regularizing effect of absorption terms in singular and degenerate elliptic problems, arXiv: 2008.03597. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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