Biblioteca de la Universidad Complutense de Madrid

Elliptic and parabolic quasilinear equations giving rise to a free boundary:the boundary of the support of the solution.


Díaz Díaz, Jesús Ildefonso (1986) Elliptic and parabolic quasilinear equations giving rise to a free boundary:the boundary of the support of the solution. Proceedings of Symposia in Pure Mathematics, 45 (1). pp. 381-393. ISSN 0082-0717

[img] PDF
Restringido a Sólo personal autorizado del repositorio hasta 31 Diciembre 2020.


URL Oficial:


This paper gives a very brief review of some properties of solutions of quasilinear scalar elliptic and parabolic partial differential equations in a spatial domain Ω. Emphasis is laid on nonlinearities which allow the support of the solutions to be smaller than Ω. The use of sub- and supersolutions and energy estimates are the only techniques described in any detail, but the paper gives the reader an entree into the vast theoretical literature on these problems. The physical references are selective

Tipo de documento:Artículo
Palabras clave:quasilinear equations; free boundary; support of the solutions; porous media equation; supersolutions
Materias:Ciencias > Matemáticas > Geometría diferencial
Código ID:16412

H.W.Alt and D.Phillips,A free boundary prohlem for semilinear elliptic equations(to appear).

S. N. Antonsev, On the localization of solutions of nonlinear degenerate elliptic and parabolic Equations, Soviet Math. Dokl. 24 (1981), 420-424.

R. Aris, The mathematical theory of diffusion and reaction impermeable catalysis, Clarendon Press,Oxford,1975.

D. G. Aronson, Regularity properties of flows through porous media, SIAM Appl. Malh. 17(1969),461-467.

D. G. Aronson, Regularity properties of flows through porous media: the interface, Arch,Rational Mech.Anal. 37 (1970), 1-10.

R. G. Aronson, Some properties of the interface for gas flow in porous media, Free Boundary Problems: Theory and Applicalions, Pitman, New York, 1983.

C. Atkinson and J. E. Bouillet, Some qualitative properties of solutions of a generalised diffusion equation, Math. Proc. Cambridge Philos. Soc. 86 (1979), 495-510.

C. Atkinson and J. E. Bouillet, A generalized dlffusion equation: Radial symmetries and camparison theorems, J. Math. Anal. Appl. 95 (1983), 37-68.

A. Bamberger,Etude d'une équations doublement non linéaire, J. Funct. Anal. 24 (1977), 148-155.

C. Bandle, R. P. Sperb and I. Stakgold, Diffusion and reaction with monotone kinetics, Nonlinear Anal. 8 (1984), 321-333.

C. Bandle and I. Stakgold, The formation of the dead core in parabolic reaction-diffusion problems,Trans. Amer. Malh. Soc. 286 (1984), 275-293.

J. Bear, Dynamics of fluids inporous media, American Elsevier, New York, 1971.

Ph. Benilan, H. Brezis and M. G. Crandall, A semilinear equation in Ll( RN), Ann. Scuola Norm.Sup. Pisa 2 (1975), 523-555.

A. Bensoussan and J. L. Lions, On the support of the solution of some variational inequalities of evolution, J. Math. Soc. Japan 28 (1976), 1-17.

M.F.Bidaut-Veron, Propriété de support compact de la solution d'une équation aux derivées partielles nonlineaires d'ordre 4, C. R. Acad. Sci. Paris 287 (1975), 1005-1008.

F. Bernis, Compactness of the support for some nonlinear elliptic problems of arbitrary order in dimension N, Comm. Parlial Differential Equations 9 (1984), 271-312.

M. Bertsch and L. A. Peletier, Porous media type equations: an overview (to appear).

M. Bertsch and L. A. Peletier, A positivity property of solutions of non-linear dlffusion equations, J.Differential Equations (to appear).

M. Bertsch, R. Kersner and L. A. Peletier, Sur le comportement de la frontiere libre dans une equation en theorie de la filtration, C. R. Acad. Sci. Paris 295(1982), 63-66.

M. Bertsch, R. Kersner, Positivity versus localization in degenerate diffusion equations (to appear).

M. Bertsch, P. de Motoni and L. A. Peletier, Degenerate difusion andmi the Stefan Problem (to appear).

J. G. Berryman and C. J. Holland, Stability of the separable soluton for fast diffusion, Arch.Rational Mech. Anal. 74 (1980), 279-288.

C. M. Brauner and B. Nikolaenko, On nonlinear eigenvalule problems wich estend into free boundaries problems, Bifurcation and Nonlinear Eigenvalue Problems (c. Bardos, J. M. Lasry and M.Sachlzman, eds.), Lecture Notes in Math., vol. 782, Springer, 1980.

H. Brezis, Solutions of variational inequalities with compact support, Uspekhi Mat. Nauk 129 (1974), 103-108,

H. Brezis and A. Friedman, Estimates in the support of solutions of parabolic variational inequalities, Illinois J. Math. 20 (1976),82-97.

E. D. Conway, The formation and decay of shocks for a conservation law in several dimensionsm, Arch. Rational Mech. Anal. 64 (1977), 47-59.

G. Díaz, Estimation de l'ensemble de coincidence de la Solutions des prpbñemes d’obstacle pour les equations de Hamilton-Jacobi-Bellman, C. R. Acad. Sci. Paris 290 (1980), 587-591.

G. Díaz, Same properties of the solutions of degenerate second order P DE in non-divergence form,Applicable Anal. (to appear).

G. Díaz and J. I. Díaz, Finite extinction time for a class of nonlinear parabolic equations, Comm.Partial Differential Equations 4 (1979), 1213-1231.

J. I. Díaz, Solutions with compact support for some degenerate parabolic problems, Nonlinear Anal.3 (1979), 831-847,

J. I. Díaz, Anulación de soluciones para operadores acretivos en espacios de Banach, Rev. Real Acad. Cienc, Exact. Fis. Natur. Madrid 74 (1980),865-880,

J. I. Díaz, Técnica de supersoluciones locales para problems estacionarios no lineales, Memoria nºXVI, Real Academia de Ciencias, Madrid, 1982.

J. I. Díaz, Nonlinear partial differential equations and free boundaries, Vol. I: Elliptic Equationls,Research Notes in Math., no. 106, Pitman, London, 1985; Vol. II: Parabolic and Hyperbolic Equations,in preparation.

J. I. Díaz and J. Hernández, On the existence of a free boundary for a class of reaction-diffusion systems, SIAM J. Math, Anal, 5 (1984), 670-685.

J. I.Díaz and J.Hernández, Somne results on the existence of free boundaries for parabolic reaction-diffusion systems, Trends in Theory and Practice of Nonlinear Differential Equations(Proc. 5th Conf. on Trends in Theory and Practice of Nonlinear Differential Equations,Arlington, Texas, 1982) V, Lakshmikantham,ed. Lecture Notes Pure Appl. Math. no. 90, Marcel Dekker, New York, 1984, pp, 149-158.

J. I. Díaz and M. A. Herrero, Proprietés de support compact pour certaines equations elliptiques et paraboliques nonlinearies, C. R. Acad. Sci. Paris 286 (1978), 815-817.

J. I. Díaz and M. A. Herrero, Estimates of the support of the solutions of some nonlinear elliptic and parabolic problems, Proc, Roy, Soc. Edinburgh Sect. A 89 (1981), 249-258.

J. I. Díaz and R. Jimenez, Boundary behaviour of solutions of Signorini type problems (to appear).

J. I. Díaz and R, Kersner, Non existence d'une des frontieres libres dans une equation degénérée en théorie de la filtration, C. R. Acad, Sci Paris 296 (1983),505-508.

J. I. Díaz and L. Veron, Existence theory and qualitative properties of the solutions of some first order quasilinear variational inequalities, Indiana Univ,Math,J,32 (1983),319-361.

J. I. Díaz and L. Veron, Compacité du support des solutions d'equations quasilinearies elliptiques ou paraboliques, C. R. Acad. Sci Paris 297 (1983),149-152.

J. I. Díaz and L. Veron, Local vanishing properties of solutions of elliptic and parabolic quasilinear equations,

Trans. Amer. Math. Soc, 289 (1985),

L, C. Evans and B. Knerr, Instantaneous shrinking of the support of non-negative solutions to certain nonlinear parabolic equations and variational inequalities, Illinois J, Math. 23 (1979), 153-166.

C. Francsis, On the porous medium equation with lower order singular nonlinear terms(to appear).

A, Friedman, Boundary behaviour of solutions of variational inequalities for elliptic operators, Arch. Rational Mech. Anal. 27 (1967), 95-107.

A, Friedman, Variational principles and free boundary problems, Wiley, New York, 1982.

A. Friedman and P. Phillips, The free bouundary of a semilinear elliptic equation Trans, Amer.Math. Soc. 282 (1984), 153-182.

V. A. Galaktionov, A boundary value problem for the nonlinear parabolic equation u = Δu +1 + u J. Differential Equations 17 (1981), 551-555.

B. H. Gilding, Properties of solutions of an equation in theory of filtration, Arch, Rational Mech. Anal. 65 (1977), 203-225.

B. H. Gilding, A nonlinear degenerate parabolic equation, Ann. Scuola Norm. Sup. Pisa 4 (1977),393-432,

M. E. Gurtin and A. C. Pipkin, A general theory of heat conduction with finite wave speeds, Arch.Rational Mech. Anal. 31 (1968), 113-[26.

M. A. Herrero and J, L. Vázquez, Asymptotic behaviour of solutions of a strongly nonlinear parabolic problem, Ann. Fac. Sci. Toulouse Math. 3 p 981), 113-127.

M. A. Herrero and J, L. Vázquez,, On the propagation properties of a nonlinear degenerate parabolic equation, Comm. Partial Differential Equations 7 (1982),1381-1402.

A. S. Kalashnikov, The propagation of disturbances in problems of nonllinear heat conduction with absorption, Zh. Vychisl. Mal. i Mat. Fiz. 14 (1974), 891-905.

A. S. Kalashnikov, On the character of the propagation of perturbations in processes described by quasilinear

degenerate parabolic equations, Trudy Sem. Petrovsky (1975),135-144.

A. S. Kalashnikov. On a nonlincar equation appearing in the theory of non-stationary filtration. Trudy Sem. Petrovsk. 4 (1978), 137-146.

A. S. Kalashnikov, The concept of a finite rate of propagation of a perturbation, Russian Math. Surveys 34


S. Kamin and Ph. Rosenau. Thermal waves in a absorbing and convecting medium (to appear).

R. Kersner, The behaviour of temperature fronts in media with nononlinear thermal conductivity under absorption, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 33 (l978), 44-51.

R. Kersner, Degenerate parabolic equations with general nonlinearities, Nonlinear Anal. 4 (19S0),1043-1061.

R. Kersner, Filtration with absorption: necesary and sufficient condition for the propagation or perturbations to have finite velocity, J. Math. Anal. Appl. (1983).

B. Knerr, The porous medium equation in one dimension, Trans. Arner. Math. Soc. 234 (1977),381-415.

R. Kersner, The behaviour of the support of solutions of the equation of nonlinear heat conduction with absorption in one dimension, Trans. Amer. Math. Soc. 249 (1979), 409-424.

S. N. Kruzkov, First order quasilinear equations in several independent variables, Math. USSR-Sb.10 (1970), 217-243.

P. D. Lax, Hyperbolic systems of conservation laws and the mathematical theory of shock waves,C.B.M.S. Regional Conf. Ser. in Applied Math., no. 11, SIAM, Philadelphia, 1973.

T. Nagai and M. Mimura, Asymptotic behaviour of a nonlinear degenerate diffusion in a equation in a population dynamics, SIAM J. Appl. Math. 43 (1983), 449-469.

O. A. Oleinik, A. S. Kalashnikov and Chzhou Yui-Lin, The Cauchy problem ans boundary problems for equations of the type of nonstationary filtration, Izv. Akad. Nauk SSSR Ser. Mal. 22 (1958),667-704.

L. A. Pcletier, A necessary and sufficient condition for the existence of an interface in flows through Porous media, Arch. Rational Mech. Anal. 56 (1974), 163-190.

R. Redheffer, On a nonlinear functional of Berkcovitz and Pollard, Arch. Rational Mech. Anal. 50 (1973),1-9.

E. S. Sabinina, A class of nonlinear degenerating parabolic equations, Soviet Math. Dokl. 3 (1962),495-498.

J. Serrin, The solvability of boundary value problems, Proc. Sympos. Pure Math., vol. 28, Amer. Math. Soc., Providence, R. l., 1976, pp. 507-524.

I. Stakgold, Estimates for some free boundary problems, Ordinary and Partial Differential Equations (W. N. Everitt and B. D. Sleeman, eds.),Lecture Notes in Math., vol. 846, Springer-Verlag,1981.

J. L. Vázquez, Asymptotic behaviour amd propagatiom properties of the one-dimensional flow of a gas ill a porous medium, Trans. Amer. Math. Soc. 277 (1983), 507-527.

J. L. Vázquez, Behaviour of the velocity of one-dimensional flows in porous media (to appear).

J. L. Vázquez, The interfaces of one-dimensional flows in porous media (to appear).

L. Veron, Eflects regularisant de semi-groupes non lineaires dans les espaces de Banach, Ann. Fac.Sci. Toulouse Math. 1(1979),171-200.

L. Veron, Equations d'evolution semi-lineaires du second ordre dans L , Rev. Roumaine Math. Pures. Appl. 27 (1982), 95-123.

Y. B. Zeldovich and Y P. Raizer, Physics of shock waves and high-temperature hydrodynamic phenomena, Academic Press, New York, 1969.

Depositado:18 Sep 2012 08:47
Última Modificación:07 Feb 2014 09:28

Sólo personal del repositorio: página de control del artículo