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 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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