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One Class of Sobolev Type Equations of Higher Order with Additive "White Noise"
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 |
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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