# American Institute of Mathematical Sciences

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.
 [1] Gisella Croce. An elliptic problem with degenerate coercivity and a singular quadratic gradient lower order term. Discrete and Continuous Dynamical Systems - S, 2012, 5 (3) : 507-530. doi: 10.3934/dcdss.2012.5.507 [2] Rosaria Di Nardo. Nonlinear parabolic equations with a lower order term and $L^1$ data. Communications on Pure and Applied Analysis, 2010, 9 (4) : 929-942. doi: 10.3934/cpaa.2010.9.929 [3] Angelo Favini, Alfredo Lorenzi, Hiroki Tanabe, Atsushi Yagi. An $L^p$-approach to singular linear parabolic equations with lower order terms. Discrete and Continuous Dynamical Systems, 2008, 22 (4) : 989-1008. doi: 10.3934/dcds.2008.22.989 [4] Lucio Boccardo, Maria Michaela Porzio. Some degenerate parabolic problems: Existence and decay properties. Discrete and Continuous Dynamical Systems - S, 2014, 7 (4) : 617-629. doi: 10.3934/dcdss.2014.7.617 [5] U. Biccari, V. Hernández-Santamaría, J. Vancostenoble. Existence and cost of boundary controls for a degenerate/singular parabolic equation. Mathematical Control and Related Fields, 2022, 12 (2) : 495-530. doi: 10.3934/mcrf.2021032 [6] Seick Kim, Longjuan Xu. Green's function for second order parabolic equations with singular lower order coefficients. Communications on Pure and Applied Analysis, 2022, 21 (1) : 1-21. doi: 10.3934/cpaa.2021164 [7] Maoji Ri, Shuibo Huang, Canyun Huang. Non-existence of solutions to some degenerate coercivity elliptic equations involving measures data. Electronic Research Archive, 2020, 28 (1) : 165-182. doi: 10.3934/era.2020011 [8] Rubén Figueroa, Rodrigo López Pouso, Jorge Rodríguez–López. Existence and multiplicity results for second-order discontinuous problems via non-ordered lower and upper solutions. Discrete and Continuous Dynamical Systems - B, 2020, 25 (2) : 617-633. doi: 10.3934/dcdsb.2019257 [9] Micol Amar, Virginia De Cicco. Lower semicontinuity for polyconvex integrals without coercivity assumptions. Evolution Equations and Control Theory, 2014, 3 (3) : 363-372. doi: 10.3934/eect.2014.3.363 [10] Judith Vancostenoble. Improved Hardy-Poincaré inequalities and sharp Carleman estimates for degenerate/singular parabolic problems. Discrete and Continuous Dynamical Systems - S, 2011, 4 (3) : 761-790. doi: 10.3934/dcdss.2011.4.761 [11] Teemu Lukkari, Mikko Parviainen. Stability of degenerate parabolic Cauchy problems. Communications on Pure and Applied Analysis, 2015, 14 (1) : 201-216. doi: 10.3934/cpaa.2015.14.201 [12] Annamaria Canino, Elisa De Giorgio, Berardino Sciunzi. Second order regularity for degenerate nonlinear elliptic equations. Discrete and Continuous Dynamical Systems, 2018, 38 (8) : 4231-4242. doi: 10.3934/dcds.2018184 [13] Lucio Boccardo. Some Dirichlet problems with bad coercivity. Discrete and Continuous Dynamical Systems, 2002, 8 (2) : 319-329. doi: 10.3934/dcds.2002.8.319 [14] Daniela Giachetti, Maria Michaela Porzio. Global existence for nonlinear parabolic equations with a damping term. Communications on Pure and Applied Analysis, 2009, 8 (3) : 923-953. doi: 10.3934/cpaa.2009.8.923 [15] Gary Lieberman. A new regularity estimate for solutions of singular parabolic equations. Conference Publications, 2005, 2005 (Special) : 605-610. doi: 10.3934/proc.2005.2005.605 [16] Changchun Liu. A fourth order nonlinear degenerate parabolic equation. Communications on Pure and Applied Analysis, 2008, 7 (3) : 617-630. doi: 10.3934/cpaa.2008.7.617 [17] Shaohua Chen. Boundedness and blowup solutions for quasilinear parabolic systems with lower order terms. Communications on Pure and Applied Analysis, 2009, 8 (2) : 587-600. doi: 10.3934/cpaa.2009.8.587 [18] Charles A. Stuart. Stability analysis for a family of degenerate semilinear parabolic problems. Discrete and Continuous Dynamical Systems, 2018, 38 (10) : 5297-5337. doi: 10.3934/dcds.2018234 [19] Lin Yan, Bin Wu. Null controllability for a class of stochastic singular parabolic equations with the convection term. Discrete and Continuous Dynamical Systems - B, 2022, 27 (6) : 3213-3240. doi: 10.3934/dcdsb.2021182 [20] Alexandre Montaru. Wellposedness and regularity for a degenerate parabolic equation arising in a model of chemotaxis with nonlinear sensitivity. Discrete and Continuous Dynamical Systems - B, 2014, 19 (1) : 231-256. doi: 10.3934/dcdsb.2014.19.231

2020 Impact Factor: 2.425

## Metrics

• PDF downloads (479)
• HTML views (480)
• Cited by (0)

## Other articlesby authors

• on AIMS
• on Google Scholar

[Back to Top]