Complutense University Library

Existence theory and qualitative properties of the solutions of some first order quasilinear variational inequalities.


Díaz Díaz, Jesús Ildefonso and Véron, Laurent (1983) Existence theory and qualitative properties of the solutions of some first order quasilinear variational inequalities. Indiana University Mathematics Journal, 32 (3). pp. 319-361. ISSN 0022-2518

[img] PDF
Restringido a Repository staff only hasta 31 December 2020.


Official URL:


The authors consider first-order equations in "conservation laws'' form perturbed by a semilinear nonlinearity of monotone type. The known existence and uniqueness results for conservation laws—results due to Kruzhkov—are easily adapted to this situation and some qualitative properties of the solutions are discussed.

Item Type:Article
Uncontrolled Keywords:entropy condition; conservation laws; multivalued operator; comparison principles; finite speed of propagation; Cauchy problem; maximal monotone graph; solution in Kruzkov sense; semigroup solution; compact support
Subjects:Sciences > Mathematics > Differential equations
ID Code:16430

PR. BENILAN, Equations d'évolutions dans un espace de Banach et applications, These d'Etat,Université d'Orsay, 1972.

PR. BENILAN, Equations quasilinéaires du premier ordre (preprint).

PH. BENILAN & M. G. CRANDALL, Regularizing effects of homogeneous evolution equations,

A. BENSOUSSAN & J. L. LIONS, Inéquations variationnelles non linéaires du premier et du second ordre, C. R. Acad. Sci. Paris 276 (1973), 1411-1415.

A. BENSOUSSAN & J. L. LIONS, On the support of the solutions of some variational inequalities of evolution, J. Math. Soc. Japan 28 (1976), 1-17.

H. BREZIS, Opérateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert, North-Holland, Amsterdam 1973.

H. BREZIS, Monotone operators, nonlinear semigroups and applications, Proc. Int. Congress Math., Vancouver, 1974.

H. BREZIS & I. EKELAND, Un principe variationnel associé a certaines équations paraboliques.Le cas indépendant du temps, C. R. Acad. Sci. Paris 282 (1976), 971-974.

H. BREZIS & A. FRIEDMAN,Estimates on the support of solutions of parabolic variational inequalities,Illinois J. Math. 20 (1976),82-97.

E. CONWAY & J. SMOLLER, Global solutions of the Cauchy problem for quasilinear first order equations in several space variables, Comm. Pure Appl. Math. 19 (1966), 95-105.

M. G. CRANDALL, The semigroup approach to first order quasilinear equations in several space variables, Israel J. Math. 12 (1972), 168-192.

M. G. CRANDALL & T. M. LIGGETT, Generation of semigroups of nonlinear transformations on general Banach spaces, Amer. J. Math. 93 (1971), 265-298.

C.M. DAFERMOS, Generalized characteristics and the structure of solutions of hyperbolic conservation laws, Indiana Univ. Math. J. 26 (1977), 1097-1119.

C. M. DAFERMOS, Asymptotic behaviour of solutions of hyperbolic balance laws in "Bifurcation Phenomena in Mathematical Physics and Related Topics,» (C. Bardos & D. Bessio, Eds.),Reidel Publishing Co. 1978.

J. I. DÍAZ, Propriedades cualitativas de ciertos problemas parabolicos no lineales, una classification para los modelos de difusion del calor, Real Acad. Cienc. Madrid XIV (1980).

J. I. DÍAZ,Tecnica de supersoluciones locales para problemas estacionarios non lineales. Applicacional estudio de flujos subsonicos, Real Acad. Cienc. Madrid XVI (1982).

N. DUNFORD & J. L. SCHWARTZ, Linear Operators, Part 1, General Theory, Interscience Publishers,Inc., New York, 1957.

L. C. EVANS, Nonlinear evolution equations in an arbitrary Banach space, Israel J. Math. 26(1977), 1-42.

L. C. EVANS & B. KNERR, lnstantaneous shrinking of the support of nonnegative solutions to certain nonlinear parabolic equations and variational inequalities, Illinois J. Math. 23 (1979),153-166.

M. A. HERRERO, Sobre el comportamiento asintotico de ciertas problemas parabolicos, Real Acad. Cienc. Madrid,(to appear).

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

P. D. LAX, Hyperbolic systems of conservation laws and the mathematical theory ofshock waves,C.B.M.S. Regional Conferences Series in Applied Math., 11, SIAM, Philadelphia, (1973).

O. A. OLEINIK, Discontinuous solutions ofnonlinear differential equations, Uspehi Math. Nauk.12 (1957), 3-13.

B. F. QUINN, Solutions with shocks: an example of an L1-contractive semigroup, Comm. Pure Appl. Math. 24 (1971), 125-132.

L. VERON, Equations d'évolution semi-linéaires du second ordre dans L1, Rev. Roumaine Math. Pures Appl. 27 (1982), 95-123.

L. VERON, Effets regularisants de semi-groupes non linéaires dans des espaces de Banoch, Ann.Fae. Sei. tou1ouse Math. 1 (1919), 111-200.

L. VERON, Some remarks on the convergence of approximate solutions of some nonlinear evolution equations in Hilbert space, Math. Comp. 39 (1982), 325-331.

L. VERON, Weak weak-star compactness of dominated subsets of L~(E; L:(F), Houston J. Math.(to appear).

A. I. VOL'PERT, The space B.V. and quasilinear equations, Math. USSR-Sb. 2 (1961), 225-267.

Deposited On:19 Sep 2012 08:11
Last Modified:07 Feb 2014 09:29

Repository Staff Only: item control page