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

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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

Dominique Blanchard ^1, , Olivier Guibé ^2, and Hicham Redwane ^3,

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

Abstract
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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 $u$.

Keywords: Nonlinear parabolic equations, uniqueness, renormalized solutions., existence.

Mathematics Subject Classification: Primary: 47A15; Secondary: 46A32, 47D2.

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 and 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, Ann. Scuola Norm. Sup. Pisa, 22 (1995), 241-273.
[2]	D. Blanchard, Truncations and monotonicity methods for parabolic equations, Nonlinear Anal., 21 (1993), 725-743. doi: 10.1016/0362-546X(93)90120-H.
[3]	D. Blanchard and G. Francfort, A few results on a class of degenerate parabolic equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 18 (1991), 213-249.
[4]	D. Blanchard and F. Murat, Renormalised solution for nonlinear parabolic problems with $L^1$ data, existence and uniqueness, Proc. Roy. Soc. Edinburgh Sect. A, 127 (1997), 1137-1152. doi: 10.1017/S0308210500026986.
[5]	D. Blanchard, F. Murat and H. Redwane, Existence and uniqueness of a renormalized solution for a fairly general class of nonlinear parabolic problems, J. Differential Equations, 177 (2001), 331-374. doi: 10.1006/jdeq.2000.4013.
[6]	D. Blanchard, F. Petitta and H. Redwane, Renormalized solutions of nonlinear parabolic equations with diffuse measure data, Manuscripta Math., 141 (2013), 601-635. doi: 10.1007/s00229-012-0585-7.
[7]	D. Blanchard and A. Porretta, Stefan problems with nonlinear diffusion and convection, J. Differential Equations, 210 (2005), 383-428. doi: 10.1016/j.jde.2004.06.012.
[8]	D. Blanchard and H. Redwane, Renormalized solutions for a class of nonlinear parabolic evolution problems, J. Math. Pures Appl, 77 (1998), 117-151. doi: 10.1016/S0021-7824(98)80067-6.
[9]	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.
[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 Recent Advances in Nonlinear Elliptic and Parabolic Problems (Nancy, 1988), vol. 208 of Pitman Res. Notes Math. Ser., Longman Sci. Tech., Harlow, 1989, 229-246.
[11]	L. Boccardo, F. Murat and J.-P. Puel, Existence of bounded solutions for nonlinear elliptic unilateral problems, Ann. Mat. Pura Appl. (4), 152 (1988), 183-196. doi: 10.1007/BF01766148.
[12]	J. Carrillo, Entropy solutions for nonlinear degenerate problems, Arch. Ration. Mech. Anal., 147 (1999), 269-361. doi: 10.1007/s002050050152.
[13]	J. Carrillo and P. Wittbold, Uniqueness of renormalized solutions of degenerate elliptic-parabolic problems, J. Differential Equations, 156 (1999), 93-121. doi: 10.1006/jdeq.1998.3597.
[14]	G. Dal Maso, F. Murat, L. Orsina and A. Prignet, Renormalized solutions of elliptic equations with general measure data, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 28 (1999), 741-808.
[15]	R. Di Nardo, F. Feo and O. Guibé, Uniqueness of renormalized solutions to nonlinear parabolic problems with lower-order terms, Proc. Roy. Soc. Edinburgh Sect. A, 143 (2013), 1185-1208. doi: 10.1017/S0308210511001831.
[16]	R.-J. DiPerna and P.-L. Lions, On the Cauchy problem for Boltzmann equations : global existence and weak stability, Ann of Math, 130 (1989), 321-366. doi: 10.2307/1971423.
[17]	J. Droniou, A. Porretta and A. Prignet, Parabolic capacity and soft measures for nonlinear equations, Potential Anal., 19 (2003), 99-161. doi: 10.1023/A:1023248531928.
[18]	J. Droniou and A. Prignet, Equivalence between entropy and renormalized solutions for parabolic equations with smooth measure data, NoDEA Nonlinear Differential Equations Appl., 14 (2007), 181-205. doi: 10.1007/s00030-007-5018-z.
[19]	P. Gwiazda, P. Wittbold, A. Wróblewska and A. Zimmermann, Renormalized solutions of nonlinear elliptic problems in generalized Orlicz spaces, J. Differential Equations, 253 (2012), 635-666. doi: 10.1016/j.jde.2012.03.025.
[20]	R. Landes, On the existence of weak solutions for quasilinear parabolic initial-boundary value problems, Proc. Roy. Soc. Edinburgh Sect. A, 89 (1981), 217-237. doi: 10.1017/S0308210500020242.
[21]	F. Murat, Soluciones renormalizadas de EDP elipticas non lineales, Technical Report R93023, Laboratoire d'Analyse Numérique, Paris VI, 1993, Cours à l'Université de Séville.
[22]	F. Murat, Equations elliptiques non linéaires avec second membre $L^1$ ou mesure, in Compte Rendus du 26ème Congrès d'Analyse Numérique, les Karellis, 1994, A12-A24.
[23]	F. Petitta, Asymptotic behavior of solutions for linear parabolic equations with general measure data, C. R. Math. Acad. Sci. Paris, 344 (2007), 571-576. doi: 10.1016/j.crma.2007.03.021.
[24]	F. Petitta, Renormalized solutions of nonlinear parabolic equations with general measure data, Ann. Mat. Pura Appl. (4), 187 (2008), 563-604. doi: 10.1007/s10231-007-0057-y.
[25]	F. Petitta, A. C. Ponce and A. Porretta, Diffuse measures and nonlinear parabolic equations, J. Evol. Equ., 11 (2011), 861-905. doi: 10.1007/s00028-011-0115-1.
[26]	A. Porretta, Existence results for nonlinear parabolic equations via strong convergence of truncations, Ann. Mat. Pura Appl. (4), 177 (1999), 143-172. doi: 10.1007/BF02505907.
[27]	A. Prignet, Existence and uniqueness of "entropy'' solutions of parabolic problems with $L^1$ data, Nonlinear Anal., 28 (1997), 1943-1954. doi: 10.1016/S0362-546X(96)00030-2.
[28]	H. Redwane, Existence of a solution for a class of parabolic equations with three unbounded nonlinearities, Adv. Dyn. Syst. Appl., 2 (2007), 241-264.
[29]	H. L. Royden, Real Analysis, Third edition, Macmillan Publishing Company, New York, 1988.
[30]	J. Serrin, Pathological solution of elliptic differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 18 (1964), 385-387.
[31]	J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pur. App, 146 (1987), 65-96. doi: 10.1007/BF01762360.

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, Ann. Scuola Norm. Sup. Pisa, 22 (1995), 241-273.
[2]	D. Blanchard, Truncations and monotonicity methods for parabolic equations, Nonlinear Anal., 21 (1993), 725-743. doi: 10.1016/0362-546X(93)90120-H.
[3]	D. Blanchard and G. Francfort, A few results on a class of degenerate parabolic equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 18 (1991), 213-249.
[4]	D. Blanchard and F. Murat, Renormalised solution for nonlinear parabolic problems with $L^1$ data, existence and uniqueness, Proc. Roy. Soc. Edinburgh Sect. A, 127 (1997), 1137-1152. doi: 10.1017/S0308210500026986.
[5]	D. Blanchard, F. Murat and H. Redwane, Existence and uniqueness of a renormalized solution for a fairly general class of nonlinear parabolic problems, J. Differential Equations, 177 (2001), 331-374. doi: 10.1006/jdeq.2000.4013.
[6]	D. Blanchard, F. Petitta and H. Redwane, Renormalized solutions of nonlinear parabolic equations with diffuse measure data, Manuscripta Math., 141 (2013), 601-635. doi: 10.1007/s00229-012-0585-7.
[7]	D. Blanchard and A. Porretta, Stefan problems with nonlinear diffusion and convection, J. Differential Equations, 210 (2005), 383-428. doi: 10.1016/j.jde.2004.06.012.
[8]	D. Blanchard and H. Redwane, Renormalized solutions for a class of nonlinear parabolic evolution problems, J. Math. Pures Appl, 77 (1998), 117-151. doi: 10.1016/S0021-7824(98)80067-6.
[9]	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.
[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 Recent Advances in Nonlinear Elliptic and Parabolic Problems (Nancy, 1988), vol. 208 of Pitman Res. Notes Math. Ser., Longman Sci. Tech., Harlow, 1989, 229-246.
[11]	L. Boccardo, F. Murat and J.-P. Puel, Existence of bounded solutions for nonlinear elliptic unilateral problems, Ann. Mat. Pura Appl. (4), 152 (1988), 183-196. doi: 10.1007/BF01766148.
[12]	J. Carrillo, Entropy solutions for nonlinear degenerate problems, Arch. Ration. Mech. Anal., 147 (1999), 269-361. doi: 10.1007/s002050050152.
[13]	J. Carrillo and P. Wittbold, Uniqueness of renormalized solutions of degenerate elliptic-parabolic problems, J. Differential Equations, 156 (1999), 93-121. doi: 10.1006/jdeq.1998.3597.
[14]	G. Dal Maso, F. Murat, L. Orsina and A. Prignet, Renormalized solutions of elliptic equations with general measure data, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 28 (1999), 741-808.
[15]	R. Di Nardo, F. Feo and O. Guibé, Uniqueness of renormalized solutions to nonlinear parabolic problems with lower-order terms, Proc. Roy. Soc. Edinburgh Sect. A, 143 (2013), 1185-1208. doi: 10.1017/S0308210511001831.
[16]	R.-J. DiPerna and P.-L. Lions, On the Cauchy problem for Boltzmann equations : global existence and weak stability, Ann of Math, 130 (1989), 321-366. doi: 10.2307/1971423.
[17]	J. Droniou, A. Porretta and A. Prignet, Parabolic capacity and soft measures for nonlinear equations, Potential Anal., 19 (2003), 99-161. doi: 10.1023/A:1023248531928.
[18]	J. Droniou and A. Prignet, Equivalence between entropy and renormalized solutions for parabolic equations with smooth measure data, NoDEA Nonlinear Differential Equations Appl., 14 (2007), 181-205. doi: 10.1007/s00030-007-5018-z.
[19]	P. Gwiazda, P. Wittbold, A. Wróblewska and A. Zimmermann, Renormalized solutions of nonlinear elliptic problems in generalized Orlicz spaces, J. Differential Equations, 253 (2012), 635-666. doi: 10.1016/j.jde.2012.03.025.
[20]	R. Landes, On the existence of weak solutions for quasilinear parabolic initial-boundary value problems, Proc. Roy. Soc. Edinburgh Sect. A, 89 (1981), 217-237. doi: 10.1017/S0308210500020242.
[21]	F. Murat, Soluciones renormalizadas de EDP elipticas non lineales, Technical Report R93023, Laboratoire d'Analyse Numérique, Paris VI, 1993, Cours à l'Université de Séville.
[22]	F. Murat, Equations elliptiques non linéaires avec second membre $L^1$ ou mesure, in Compte Rendus du 26ème Congrès d'Analyse Numérique, les Karellis, 1994, A12-A24.
[23]	F. Petitta, Asymptotic behavior of solutions for linear parabolic equations with general measure data, C. R. Math. Acad. Sci. Paris, 344 (2007), 571-576. doi: 10.1016/j.crma.2007.03.021.
[24]	F. Petitta, Renormalized solutions of nonlinear parabolic equations with general measure data, Ann. Mat. Pura Appl. (4), 187 (2008), 563-604. doi: 10.1007/s10231-007-0057-y.
[25]	F. Petitta, A. C. Ponce and A. Porretta, Diffuse measures and nonlinear parabolic equations, J. Evol. Equ., 11 (2011), 861-905. doi: 10.1007/s00028-011-0115-1.
[26]	A. Porretta, Existence results for nonlinear parabolic equations via strong convergence of truncations, Ann. Mat. Pura Appl. (4), 177 (1999), 143-172. doi: 10.1007/BF02505907.
[27]	A. Prignet, Existence and uniqueness of "entropy'' solutions of parabolic problems with $L^1$ data, Nonlinear Anal., 28 (1997), 1943-1954. doi: 10.1016/S0362-546X(96)00030-2.
[28]	H. Redwane, Existence of a solution for a class of parabolic equations with three unbounded nonlinearities, Adv. Dyn. Syst. Appl., 2 (2007), 241-264.
[29]	H. L. Royden, Real Analysis, Third edition, Macmillan Publishing Company, New York, 1988.
[30]	J. Serrin, Pathological solution of elliptic differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 18 (1964), 385-387.
[31]	J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pur. App, 146 (1987), 65-96. doi: 10.1007/BF01762360.

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