# American Institute of Mathematical Sciences

## 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

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
 $0<\gamma<1.$
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

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##### 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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