Existence of a renormalized solution of nonlinear parabolic equations with lower order term and general measure data
Resumen
We give an existence result of a renormalized solution for a class of nonlinear parabolic equations
@b(u)/@t
div(a(x; t;grad(u))+ H(x; t;ru) = ,
where the right side is a general measure, b is a strictly increasing C1-function,
div(a(x; t;grad(u)) is a Leray{Lions type operator with growth in grad(u) and H(x; t;grad(u) is a nonlinear lower order term which satisfy the growth condition with respect to grad(u).
Descargas
Citas
K. Ammar, P. Wittbold, Existence of renormalized solutions of degenerate elliptic-parabolic problems, Proc. Roy. Soc. Edinburgh Sect. A 133 (2003), 477-496.
L. Boccardo, A. Dall’Aglio, T. Gallouet, L. Orsina, Nonlinear parabolic equations with measure data, J. Funct. Anal., 147, no.1 (1997), 237-258.
L. Boccardo, T. Gallouet, Nonlinear elliptic and parabolic equations involving measure data, J. Funct. Anal., 87 (1989), 149-169.
H. Brezis, A. C. Ponce, Reduced measures for obstacle problems, Adv. Diff. Eq. 10 (2005), 1201-1234.
D. Blanchard, F. Petitta, H. Redwane, Renormalized solutions of nonlinear parabolic equations with diffuse measure data, Manuscripta Math., 141 (2013), 601-635.
M. F. Bidaut-Veron, H. N. Quoc, Stability properties for quasilinear parabolic equations with measure data and applications, preprint, arXiv:1310.5253v2.
J. Carrillo, P. Wittbold, Uniqueness of renormalized solutions of degenerate elliptic-parabolic problems, J. Differential Equations, 156 (1999), 93-121.
R. J. Di Perna, and P. L. Lions, On the Cauchy problem for Boltzmann equations: Global existence and weak stability, Ann. Math., 130 (1989), 321-366.
G. Dal Maso, F. Murat, L. Orsina, A. Prignet, Renormalized solutions of elliptic equations with general measure data, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 28, no. 4 (1999), 741-808.
J. Droniou, A. Prignet, Equivalence between entropy and renormalized solutions for parabolic equations with smooth measure data”, NoDEA Nonlinear Differential Equations Appl. 14, no. 1-2 (2007), 181-205.
J. Droniou, A.Porretta, A. Prignet, Parabolic capacity and soft measure for nonlinear equations, Potential Anal., 19, no.2 (2003), 99-161.
R. Di Nardo, Nonlinear parabolic equations with a lower order term and L1 data July 2010 Communications on Pure and Applied Analysis 9(4):929-942.
R. Di Nardo, F. Feo, O. Guib´e, Existence result for nonlinear parabolic equations with lower order terms. Anal. Appl, Singap., 9(2) (2011), 161–186.
M. Fukushima, K. Sato, S. Taniguchi, On the closable part of pre-Dirichlet forms and the fine supports of underlying measures,” Osaka J. Math., 28, (1991), 517-535.
J. L. Lions, Quelques m´ethodes de r´esolution des probl`emes aux limites non lineaire, Dunod et Gauthier-Villars, Paris, (1969).
J. Leray, J. L. Lions, Quelques r´esultats de Visik sur les problemes elliptiques non lineaires par les m´ethodes de Minty-Browder, Bull. Soc. Math. France 93 (1965), 97-107.
R. Landes, On the existence of weak solutions for quasilinear parabolic initial-boundary value problems, Proc. Roy. Soc. Edinburgh Sect., A89 (1981), 217-237.
G. G. Lorentz. Some new functional spaces. Ann. of Math. (2), 51 (1950), 37-55.
R. O’Neil, Integral transforms and tensor products on Orlicz spaces and L(p, q ) spaces. J. Analyse Math., 21 (1981), 1-276.
Prignet, A. Remarks on existence and uniqueness of solutions of elliptic problems with righthand side measures. (English, Italian summary) Rend. Mat. Appl. (7) 15 (1995), no. 3, 321337.
M. Pierre; Parabolic capacity and Sobolev spaces, SIAM J. Math. Anal. 14 (1983), 522–533.
F. Petitta, Renormalized solutions of nonlinear parabolic equations with general measure data, Ann. Mat. Pura ed Appl., 187 (4) (2008), 563-604.
M. M. Porzio. Existence of solutions for some “noncoercive” parabolic equations. Discrete Contin. Dynam. Systems, 5(3) (1999), 553-568.
F. Petitta, A. Porretta, On the notion of renormalized solution to nonlinear parabolic equations with general measure data, J. of Elliptic and Parabolic Equations, (2015), 201-214.
F. Petitta, A. C. Ponce, A. Porretta, Diffuse measures and nonlinear parabolic equations, Journal of Evolution Equations”, 11, no. 4 (2011), 861-905.
J. Simon, Compact sets in Lp(0, T;B), Ann. Mat. Pura Appl., 146 (1987) 65-96.
J. Serrin, Pathological solution of elliptic differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 18 (1987), 385-387.
Derechos de autor 2020 Boletim da Sociedade Paranaense de Matemática
Esta obra está bajo licencia internacional Creative Commons Reconocimiento 4.0.
When the manuscript is accepted for publication, the authors agree automatically to transfer the copyright to the (SPM).
The journal utilize the Creative Common Attribution (CC-BY 4.0).
