# American Institute of Mathematical Sciences

January  2016, 15(1): 197-217. doi: 10.3934/cpaa.2016.15.197

## Existence and uniqueness of a solution for a class of parabolic equations with two unbounded nonlinearities

 1 Laboratoire de Mathématiques Raphaël Salem, UMR CNRS 6085, Université de Rouen, Avenue de l'université, BP12, 76801 Saint Étienne du Rouvray cedex 2 Laboratoire de Mathématiques Raphaël Salem, UMR 6085 CNRS - Université de Rouen, Avenue de l'Université, BP.12, 76801 Saint-Étienne du Rouvray 3 Faculté des Sciences Juridiques, Économiques et Sociales, Université Hassan 1, B.P. 764. Settat. Morocco, France

Received  December 2014 Revised  April 2015 Published  December 2015

In this paper we prove the existence and uniqueness of a renormalized solution for nonlinear parabolic equations whose model is \begin{eqnarray} \frac{\partial b(u)}{\partial t} - div\big(a(x,t,u,\nabla u)\big)=f+ div (g), \end{eqnarray} where the right side belongs to $L^{1}(Q)+L^{p'}(0,T;W^{-1,p'}(\Omega))$, where $b(u)$ is a real function of $u$ and where $-div(a(x,t,u,\nabla u))$ is a Leray-Lions type operator with growth $|\nabla u|^{p-1}$ in $\nabla u$, but without any growth assumption on $u$.
Citation: Dominique Blanchard, Olivier Guibé, Hicham Redwane. Existence and uniqueness of a solution for a class of parabolic equations with two unbounded nonlinearities. Communications on Pure & Applied Analysis, 2016, 15 (1) : 197-217. doi: 10.3934/cpaa.2016.15.197
##### References:
 [1] P. Bénilan, L. Boccardo, T. Gallouët, R. Gariepy, M. Pierre and J. Vazquez, An $L^1$-theory of existence and uniqueness of solutions of nonlinear elliptic equations,, \emph{Ann. Scuola Norm. Sup. Pisa}, 22 (1995), 241. Google Scholar [2] D. Blanchard, Truncations and monotonicity methods for parabolic equations,, \emph{Nonlinear Anal.}, 21 (1993), 725. doi: 10.1016/0362-546X(93)90120-H. Google Scholar [3] D. Blanchard and G. Francfort, A few results on a class of degenerate parabolic equations,, \emph{Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)}, 18 (1991), 213. Google Scholar [4] D. Blanchard and F. Murat, Renormalised solution for nonlinear parabolic problems with $L^1$ data, existence and uniqueness,, \emph{Proc. Roy. Soc. Edinburgh Sect. A}, 127 (1997), 1137. doi: 10.1017/S0308210500026986. Google Scholar [5] D. Blanchard, F. Murat and H. Redwane, Existence and uniqueness of a renormalized solution for a fairly general class of nonlinear parabolic problems,, \emph{J. Differential Equations}, 177 (2001), 331. doi: 10.1006/jdeq.2000.4013. Google Scholar [6] D. Blanchard, F. Petitta and H. Redwane, Renormalized solutions of nonlinear parabolic equations with diffuse measure data,, \emph{Manuscripta Math.}, 141 (2013), 601. doi: 10.1007/s00229-012-0585-7. Google Scholar [7] D. Blanchard and A. Porretta, Stefan problems with nonlinear diffusion and convection,, \emph{J. Differential Equations}, 210 (2005), 383. doi: 10.1016/j.jde.2004.06.012. Google Scholar [8] D. Blanchard and H. Redwane, Renormalized solutions for a class of nonlinear parabolic evolution problems,, \emph{J. Math. Pures Appl}, 77 (1998), 117. doi: 10.1016/S0021-7824(98)80067-6. Google Scholar [9] L. Boccardo, A. Dall'Aglio, T. Gallouët and L. Orsina, Nonlinear parabolic equations with measure data,, \emph{J. Funct. Anal.}, 147 (1997), 237. doi: 10.1006/jfan.1996.3040. Google Scholar [10] L. Boccardo, J. I. Diaz, D. Giachetti and F. Murat, Existence of a solution for a weaker form of a nonlinear elliptic equation,, in \emph{Recent Advances in Nonlinear Elliptic and Parabolic Problems (Nancy, (1988), 229. Google Scholar [11] L. Boccardo, F. Murat and J.-P. Puel, Existence of bounded solutions for nonlinear elliptic unilateral problems,, \emph{Ann. Mat. Pura Appl. (4)}, 152 (1988), 183. doi: 10.1007/BF01766148. Google Scholar [12] J. Carrillo, Entropy solutions for nonlinear degenerate problems,, \emph{Arch. Ration. Mech. Anal.}, 147 (1999), 269. doi: 10.1007/s002050050152. Google Scholar [13] J. Carrillo and P. Wittbold, Uniqueness of renormalized solutions of degenerate elliptic-parabolic problems,, \emph{J. Differential Equations}, 156 (1999), 93. doi: 10.1006/jdeq.1998.3597. Google Scholar [14] G. Dal Maso, F. Murat, L. Orsina and A. Prignet, Renormalized solutions of elliptic equations with general measure data,, \emph{Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)}, 28 (1999), 741. Google Scholar [15] R. Di Nardo, F. Feo and O. Guibé, Uniqueness of renormalized solutions to nonlinear parabolic problems with lower-order terms,, \emph{Proc. Roy. Soc. Edinburgh Sect. A}, 143 (2013), 1185. doi: 10.1017/S0308210511001831. Google Scholar [16] R.-J. DiPerna and P.-L. Lions, On the Cauchy problem for Boltzmann equations : global existence and weak stability,, \emph{Ann of Math}, 130 (1989), 321. doi: 10.2307/1971423. Google Scholar [17] J. Droniou, A. Porretta and A. Prignet, Parabolic capacity and soft measures for nonlinear equations,, \emph{Potential Anal.}, 19 (2003), 99. doi: 10.1023/A:1023248531928. Google Scholar [18] J. Droniou and A. Prignet, Equivalence between entropy and renormalized solutions for parabolic equations with smooth measure data,, \emph{NoDEA Nonlinear Differential Equations Appl.}, 14 (2007), 181. doi: 10.1007/s00030-007-5018-z. Google Scholar [19] P. Gwiazda, P. Wittbold, A. Wróblewska and A. Zimmermann, Renormalized solutions of nonlinear elliptic problems in generalized Orlicz spaces,, \emph{J. Differential Equations}, 253 (2012), 635. doi: 10.1016/j.jde.2012.03.025. Google Scholar [20] R. Landes, On the existence of weak solutions for quasilinear parabolic initial-boundary value problems,, \emph{Proc. Roy. Soc. Edinburgh Sect. A}, 89 (1981), 217. doi: 10.1017/S0308210500020242. Google Scholar [21] F. Murat, Soluciones renormalizadas de EDP elipticas non lineales,, Technical Report R93023, (1993). Google Scholar [22] F. Murat, Equations elliptiques non linéaires avec second membre $L^1$ ou mesure,, in \emph{Compte Rendus du 26\eme Congr\es d'Analyse Num\'erique}, (1994). Google Scholar [23] F. Petitta, Asymptotic behavior of solutions for linear parabolic equations with general measure data,, \emph{C. R. Math. Acad. Sci. Paris}, 344 (2007), 571. doi: 10.1016/j.crma.2007.03.021. Google Scholar [24] F. Petitta, Renormalized solutions of nonlinear parabolic equations with general measure data,, \emph{Ann. Mat. Pura Appl. (4)}, 187 (2008), 563. doi: 10.1007/s10231-007-0057-y. Google Scholar [25] F. Petitta, A. C. Ponce and A. Porretta, Diffuse measures and nonlinear parabolic equations,, \emph{J. Evol. Equ.}, 11 (2011), 861. doi: 10.1007/s00028-011-0115-1. Google Scholar [26] A. Porretta, Existence results for nonlinear parabolic equations via strong convergence of truncations,, \emph{Ann. Mat. Pura Appl. (4)}, 177 (1999), 143. doi: 10.1007/BF02505907. Google Scholar [27] A. Prignet, Existence and uniqueness of "entropy'' solutions of parabolic problems with $L^1$ data,, \emph{Nonlinear Anal.}, 28 (1997), 1943. doi: 10.1016/S0362-546X(96)00030-2. Google Scholar [28] H. Redwane, Existence of a solution for a class of parabolic equations with three unbounded nonlinearities,, \emph{Adv. Dyn. Syst. Appl.}, 2 (2007), 241. Google Scholar [29] H. L. Royden, Real Analysis,, Third edition, (1988). Google Scholar [30] J. Serrin, Pathological solution of elliptic differential equations,, \emph{Ann. Scuola Norm. Sup. Pisa Cl. Sci.}, 18 (1964), 385. Google Scholar [31] J. Simon, Compact sets in the space $L^p(0,T;B)$,, \emph{Ann. Mat. Pur. App}, 146 (1987), 65. doi: 10.1007/BF01762360. Google Scholar

show all references

##### References:
 [1] P. Bénilan, L. Boccardo, T. Gallouët, R. Gariepy, M. Pierre and J. Vazquez, An $L^1$-theory of existence and uniqueness of solutions of nonlinear elliptic equations,, \emph{Ann. Scuola Norm. Sup. Pisa}, 22 (1995), 241. Google Scholar [2] D. Blanchard, Truncations and monotonicity methods for parabolic equations,, \emph{Nonlinear Anal.}, 21 (1993), 725. doi: 10.1016/0362-546X(93)90120-H. Google Scholar [3] D. Blanchard and G. Francfort, A few results on a class of degenerate parabolic equations,, \emph{Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)}, 18 (1991), 213. Google Scholar [4] D. Blanchard and F. Murat, Renormalised solution for nonlinear parabolic problems with $L^1$ data, existence and uniqueness,, \emph{Proc. Roy. Soc. Edinburgh Sect. A}, 127 (1997), 1137. doi: 10.1017/S0308210500026986. Google Scholar [5] D. Blanchard, F. Murat and H. Redwane, Existence and uniqueness of a renormalized solution for a fairly general class of nonlinear parabolic problems,, \emph{J. Differential Equations}, 177 (2001), 331. doi: 10.1006/jdeq.2000.4013. Google Scholar [6] D. Blanchard, F. Petitta and H. Redwane, Renormalized solutions of nonlinear parabolic equations with diffuse measure data,, \emph{Manuscripta Math.}, 141 (2013), 601. doi: 10.1007/s00229-012-0585-7. Google Scholar [7] D. Blanchard and A. Porretta, Stefan problems with nonlinear diffusion and convection,, \emph{J. Differential Equations}, 210 (2005), 383. doi: 10.1016/j.jde.2004.06.012. Google Scholar [8] D. Blanchard and H. Redwane, Renormalized solutions for a class of nonlinear parabolic evolution problems,, \emph{J. Math. Pures Appl}, 77 (1998), 117. doi: 10.1016/S0021-7824(98)80067-6. Google Scholar [9] L. Boccardo, A. Dall'Aglio, T. Gallouët and L. Orsina, Nonlinear parabolic equations with measure data,, \emph{J. Funct. Anal.}, 147 (1997), 237. doi: 10.1006/jfan.1996.3040. Google Scholar [10] L. Boccardo, J. I. Diaz, D. Giachetti and F. Murat, Existence of a solution for a weaker form of a nonlinear elliptic equation,, in \emph{Recent Advances in Nonlinear Elliptic and Parabolic Problems (Nancy, (1988), 229. Google Scholar [11] L. Boccardo, F. Murat and J.-P. Puel, Existence of bounded solutions for nonlinear elliptic unilateral problems,, \emph{Ann. Mat. Pura Appl. (4)}, 152 (1988), 183. doi: 10.1007/BF01766148. Google Scholar [12] J. Carrillo, Entropy solutions for nonlinear degenerate problems,, \emph{Arch. Ration. Mech. Anal.}, 147 (1999), 269. doi: 10.1007/s002050050152. Google Scholar [13] J. Carrillo and P. Wittbold, Uniqueness of renormalized solutions of degenerate elliptic-parabolic problems,, \emph{J. Differential Equations}, 156 (1999), 93. doi: 10.1006/jdeq.1998.3597. Google Scholar [14] G. Dal Maso, F. Murat, L. Orsina and A. Prignet, Renormalized solutions of elliptic equations with general measure data,, \emph{Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)}, 28 (1999), 741. Google Scholar [15] R. Di Nardo, F. Feo and O. Guibé, Uniqueness of renormalized solutions to nonlinear parabolic problems with lower-order terms,, \emph{Proc. Roy. Soc. Edinburgh Sect. A}, 143 (2013), 1185. doi: 10.1017/S0308210511001831. Google Scholar [16] R.-J. DiPerna and P.-L. Lions, On the Cauchy problem for Boltzmann equations : global existence and weak stability,, \emph{Ann of Math}, 130 (1989), 321. doi: 10.2307/1971423. Google Scholar [17] J. Droniou, A. Porretta and A. Prignet, Parabolic capacity and soft measures for nonlinear equations,, \emph{Potential Anal.}, 19 (2003), 99. doi: 10.1023/A:1023248531928. Google Scholar [18] J. Droniou and A. Prignet, Equivalence between entropy and renormalized solutions for parabolic equations with smooth measure data,, \emph{NoDEA Nonlinear Differential Equations Appl.}, 14 (2007), 181. doi: 10.1007/s00030-007-5018-z. Google Scholar [19] P. Gwiazda, P. Wittbold, A. Wróblewska and A. Zimmermann, Renormalized solutions of nonlinear elliptic problems in generalized Orlicz spaces,, \emph{J. Differential Equations}, 253 (2012), 635. doi: 10.1016/j.jde.2012.03.025. Google Scholar [20] R. Landes, On the existence of weak solutions for quasilinear parabolic initial-boundary value problems,, \emph{Proc. Roy. Soc. Edinburgh Sect. A}, 89 (1981), 217. doi: 10.1017/S0308210500020242. Google Scholar [21] F. Murat, Soluciones renormalizadas de EDP elipticas non lineales,, Technical Report R93023, (1993). Google Scholar [22] F. Murat, Equations elliptiques non linéaires avec second membre $L^1$ ou mesure,, in \emph{Compte Rendus du 26\eme Congr\es d'Analyse Num\'erique}, (1994). Google Scholar [23] F. Petitta, Asymptotic behavior of solutions for linear parabolic equations with general measure data,, \emph{C. R. Math. Acad. Sci. Paris}, 344 (2007), 571. doi: 10.1016/j.crma.2007.03.021. Google Scholar [24] F. Petitta, Renormalized solutions of nonlinear parabolic equations with general measure data,, \emph{Ann. Mat. Pura Appl. (4)}, 187 (2008), 563. doi: 10.1007/s10231-007-0057-y. Google Scholar [25] F. Petitta, A. C. Ponce and A. Porretta, Diffuse measures and nonlinear parabolic equations,, \emph{J. Evol. Equ.}, 11 (2011), 861. doi: 10.1007/s00028-011-0115-1. Google Scholar [26] A. Porretta, Existence results for nonlinear parabolic equations via strong convergence of truncations,, \emph{Ann. Mat. Pura Appl. (4)}, 177 (1999), 143. doi: 10.1007/BF02505907. Google Scholar [27] A. Prignet, Existence and uniqueness of "entropy'' solutions of parabolic problems with $L^1$ data,, \emph{Nonlinear Anal.}, 28 (1997), 1943. doi: 10.1016/S0362-546X(96)00030-2. Google Scholar [28] H. Redwane, Existence of a solution for a class of parabolic equations with three unbounded nonlinearities,, \emph{Adv. Dyn. Syst. Appl.}, 2 (2007), 241. Google Scholar [29] H. L. Royden, Real Analysis,, Third edition, (1988). Google Scholar [30] J. Serrin, Pathological solution of elliptic differential equations,, \emph{Ann. Scuola Norm. Sup. Pisa Cl. Sci.}, 18 (1964), 385. Google Scholar [31] J. Simon, Compact sets in the space $L^p(0,T;B)$,, \emph{Ann. Mat. Pur. App}, 146 (1987), 65. doi: 10.1007/BF01762360. Google Scholar
 [1] Peiying Chen. Existence and uniqueness of weak solutions for a class of nonlinear parabolic equations. Electronic Research Announcements, 2017, 24: 38-52. doi: 10.3934/era.2017.24.005 [2] Hiroshi Watanabe. Existence and uniqueness of entropy solutions to strongly degenerate parabolic equations with discontinuous coefficients. Conference Publications, 2013, 2013 (special) : 781-790. doi: 10.3934/proc.2013.2013.781 [3] Kenneth Hvistendahl Karlsen, Nils Henrik Risebro. On the uniqueness and stability of entropy solutions of nonlinear degenerate parabolic equations with rough coefficients. Discrete & Continuous Dynamical Systems - A, 2003, 9 (5) : 1081-1104. doi: 10.3934/dcds.2003.9.1081 [4] H. Gajewski, I. V. Skrypnik. To the uniqueness problem for nonlinear parabolic equations. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 315-336. doi: 10.3934/dcds.2004.10.315 [5] Zhigang Wang, Lei Wang, Yachun Li. Renormalized entropy solutions for degenerate parabolic-hyperbolic equations with time-space dependent coefficients. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1163-1182. doi: 10.3934/cpaa.2013.12.1163 [6] Stanislav Antontsev, Michel Chipot, Sergey Shmarev. Uniqueness and comparison theorems for solutions of doubly nonlinear parabolic equations with nonstandard growth conditions. Communications on Pure & Applied Analysis, 2013, 12 (4) : 1527-1546. doi: 10.3934/cpaa.2013.12.1527 [7] Daniela Giachetti, Maria Michaela Porzio. Global existence for nonlinear parabolic equations with a damping term. Communications on Pure & Applied Analysis, 2009, 8 (3) : 923-953. doi: 10.3934/cpaa.2009.8.923 [8] Yucheng Bu, Yujun Dong. Existence of solutions for nonlinear operator equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4429-4441. doi: 10.3934/dcds.2019180 [9] Gabriele Bonanno, Pasquale Candito, Roberto Livrea, Nikolaos S. Papageorgiou. Existence, nonexistence and uniqueness of positive solutions for nonlinear eigenvalue problems. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1169-1188. doi: 10.3934/cpaa.2017057 [10] Robert Jensen, Andrzej Świech. Uniqueness and existence of maximal and minimal solutions of fully nonlinear elliptic PDE. Communications on Pure & Applied Analysis, 2005, 4 (1) : 199-207. doi: 10.3934/cpaa.2005.4.187 [11] Tonny Paul, A. Anguraj. Existence and uniqueness of nonlinear impulsive integro-differential equations. Discrete & Continuous Dynamical Systems - B, 2006, 6 (5) : 1191-1198. doi: 10.3934/dcdsb.2006.6.1191 [12] Peter Poláčik. On uniqueness of positive entire solutions and other properties of linear parabolic equations. Discrete & Continuous Dynamical Systems - A, 2005, 12 (1) : 13-26. doi: 10.3934/dcds.2005.12.13 [13] Maria Michaela Porzio. Existence of solutions for some "noncoercive" parabolic equations. Discrete & Continuous Dynamical Systems - A, 1999, 5 (3) : 553-568. doi: 10.3934/dcds.1999.5.553 [14] T. Tachim Medjo. Existence and uniqueness of strong periodic solutions of the primitive equations of the ocean. Discrete & Continuous Dynamical Systems - A, 2010, 26 (4) : 1491-1508. doi: 10.3934/dcds.2010.26.1491 [15] Anne Mund, Christina Kuttler, Judith Pérez-Velázquez. Existence and uniqueness of solutions to a family of semi-linear parabolic systems using coupled upper-lower solutions. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-13. doi: 10.3934/dcdsb.2019102 [16] Zijuan Wen, Meng Fan, Asim M. Asiri, Ebraheem O. Alzahrani, Mohamed M. El-Dessoky, Yang Kuang. Global existence and uniqueness of classical solutions for a generalized quasilinear parabolic equation with application to a glioblastoma growth model. Mathematical Biosciences & Engineering, 2017, 14 (2) : 407-420. doi: 10.3934/mbe.2017025 [17] F. R. Guarguaglini, R. Natalini. Global existence and uniqueness of solutions for multidimensional weakly parabolic systems arising in chemistry and biology. Communications on Pure & Applied Analysis, 2007, 6 (1) : 287-309. doi: 10.3934/cpaa.2007.6.287 [18] Juncheng Wei, Wei Yao. Uniqueness of positive solutions to some coupled nonlinear Schrödinger equations. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1003-1011. doi: 10.3934/cpaa.2012.11.1003 [19] Antonio Vitolo, Maria E. Amendola, Giulio Galise. On the uniqueness of blow-up solutions of fully nonlinear elliptic equations. Conference Publications, 2013, 2013 (special) : 771-780. doi: 10.3934/proc.2013.2013.771 [20] Mikhail D. Surnachev, Vasily V. Zhikov. On existence and uniqueness classes for the Cauchy problem for parabolic equations of the p-Laplace type. Communications on Pure & Applied Analysis, 2013, 12 (4) : 1783-1812. doi: 10.3934/cpaa.2013.12.1783

2018 Impact Factor: 0.925