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Abstract results on the finite extinction time property: application to a singular parabolic equation

Díaz Díaz, Jesús Ildefonso and Belaud, Yves (2010) Abstract results on the finite extinction time property: application to a singular parabolic equation. Journal of convex analysis, 17 (3-4). pp. 827-860. ISSN 0944-6532

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Abstract

We start by studying the finite extinction time for solutions of the abstract Cauchy problem u(t) + Au + Bu = 0 where A is a maximal monotone operator and B is a positive operator on a Hilbert space H. We use a suitable spectral energy method to get some sufficient conditions which guarantee this property. As application we consider a singular semilinear parabolic equation: Au = -Delta u, Bu = a(x)u(q), a(x) >= 0 bounded and -1 < q < 1, on a regular bounded domain Omega and Dirichlet boundary conditions.

Item Type:Article
Uncontrolled Keywords:Finite extinction time, abstract Cauchy problems, singular semilinear parabolic equations, semi-classical analysis. free-boundary solutions; arbitrary order; quasilinear equations; vanishing properties; elliptic problems; energy solutions; dimension; evolution; supports
Subjects:Sciences > Physics > Mathematical physics
Sciences > Mathematics > Differential equations
ID Code:15065
References:

N. Alikakos, R. Rostamian: Lower bound estimates and separable solutions for homogeneous equations of evolution in Banach space, J. Differ. Equations 43 (1982) 323–344.

S. N. Antontsev, J. I. Díaz, S. Shmarev: Energy Methods for Free Boundary Problems, Birkhäuser, Basel (2001). H. Attouch: Équations dévolution multivoques en dimension infinie, C. R. Acad. Sci., Paris, Sér. A 274 (1972) 1289–1291

H. Attouch: On the maximality of the sum of two maximal monotone operators, Nonlinear Anal. 5(2) (1981) 143–147.

H. Attouch, H. Brezis: Duality for the sum of convex functions in general Banach spaces, in: Aspects of Mathematics and its Applications, J. A. Barroso (ed.), North-Holland, Amsterdam (1986) 125–133.

H. Attouch, A. Damlamian: Application des méthodes de convexité et monotonie à l'étude de certaines équations quasi linéaires, Proc. R. Soc. Edinb., Sect. A 79 (1977) 107–129.

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

Y. Belaud: Time-vanishing properties of solutions of some degenerate parabolic equations with strong absorption, Adv. Nonlinear Stud. 1(2) (2001) 117–152.

Y. Belaud: Asymptotic estimates for a variational problem involving a quasilinear operator in the semi-classical limit, Ann. Global Anal. Geom. 26 (2004) 271–313.

Y. Belaud, J. I. Díaz: On the finite extinction time of solutions of abstract Cauchy problems in Banach spaces, to appear.

Y. Belaud, B. Helffer, L. Véron: Long-time vanishing properties of solutions of some semilinear parabolic equations, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 18(1) (2001) 43–68.

Y. Belaud, A. E. Shishkov: Long-time extinction of solutions of some sermilinear parabolic equations, J. Differ. Equations 238 (2007) 64–86.

F. Bernis: Compactness of the support for some nonlinear elliptic problems of arbitrary order in dimension N, Commun. Partial Differ. Equations 9(3) (1984) 271–312.

J. G. Berryman, J. G. Holland: Nonlinear diffusion problems arising in plasma physics, Phys. Rev. Lett. 40 (1978) 1720–1722.

H. Brezis: Monotonicity methods in Hilbert spaces and some applications to nonlinear partial differential equations, in: Contributions to Nonlinear Functional Analysis (Madison, 1971), E. H. Zarantonello (ed.), Math. Res. Center, Univ. Wisconsin, Academic Press, New York (1971) 101–156.

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, in: Proc. Int. Congress of Mathematicians (Vancouver, B. C., 1974), Vol. 2, Canad. Math. Congress, Montréal (1975) 249–255

H. Brezis: Analyse Fonctionnelle. Théorie et Applications, Masson, Paris (1986).

J. Dávila, M. Montenegro: Positive versus free boundary solutions to a singular equation, J. Anal. Math. 90 (2003) 303–335.

G. Díaz, J. I. Díaz: Finite extinction time for a class of non linear parabolic equations, Commun. Partial Differ. Equations 4(11) (1979) 1213–1231.

J. I. Díaz: Nonlinear Partial Differential Equations and Free Boundaries, Pitman, London (1985).

J. I. Díaz: Special finite time extinction in nonlinear evolution systems: dynamic boundary conditions and Coulomb friction type problems, in: Nonlinear Elliptic and Parabolic Problems (Zürich, 2004), M. Chipot, J. Escher (eds.), Birkhäuser, Basel (2005) 71–97.

J. I. Díaz: On the Haïm Brezis pioneering contributions on the location of free boundaries, in: Elliptic and Parabolic Problems (Gaeta, 2004), M. Chipot et al. (ed.), Birkhäuser, Basel (2005) 217–234.

J. I. Díaz, J. Hernandez: Global bifurcation and continua of nonnegative solutions for a quasilinear elliptic problem, C. R. Acad. Sci., Paris, Sér. I, Math. 329 (1999) 587–592.

J. I. Díaz, J. Hernández, F. J. Mancebo: Branches of positive and free boundary solutions for some singular quasilinear elliptic problems, J. Math. Anal. Appl. 352 (2009) 449–474.

J. I. Díaz, V. Millot: Coulomb friction and oscillation: stabilization in finite time for a system of damped oscillators, CD-Rom Actas XVIII CEDYA / VIII CMA, Servicio de Publicaciones de la Univ. de Tarragona (2003).

J. I. Díaz, L. Véron: Local vanishing properties of solutions of elliptic and parabolic quasilinear equations, Trans. Amer. Math. Soc 290(2) (1985) 787–814.

D. Gilbarg, N. Trudinger: Elliptic Partial Differential Equations of Second Order, Springer, Berlin (1977).

B. Helffer: Semi-Classical Analysis for the Schrödinger Operator and Applications, Lecture Notes in Math. 1336, Springer, Berlin (1989).

M. Herrero, J. L. Vázquez: Asymptotic behaviour of the solutions of a strongly nonlinear parabolic problem, Ann. Fac. Sci. Toulouse, V. Ser, Math. 3 (1981) 113–127.

V. A. Kondratiev, L. Véron: Asymptotic behaviour of solutions of some nonlinear parabolic or elliptic equations, Asymptotic Anal. 14 (1997) 117–156.

J. Mossino: Inégalités Isopérimétriques et Applications en Physique, Travaux en Cours, Hermann, Paris (1984).

D. Phillips: Existence of solutions of quenching problems, Appl. Anal. 24 (1987) 253–264.

A. E. Shishkov, A. G. Shchelkov: Dynamics of the support of energy solutions of mixed problems for quasi-linear parabolic equations fo arbitrary order, Izv. Math. 62(3) (1998) 601–626.

A. E. Shishkov: Dead cores and instantaneous compactification of the supports of energy solutions of quasilinear parabolic equations of arbitrary order, Sb. Math. 190(12) (1999) 1843–1869.

L. Véron: Coercivité et propriétés régularisantes des semi-groupes non linéaires dans les espaces de Banach, Publication de l'Université François Rabelais - Tours (1976).

L. Véron: Effets régularisants de semi-groupes non linéaires dans des espaces de Banach, Ann. Fac. Sci. Toulouse, V. Ser, Math. 1 (1979) 171–200.

L. Véron: Singularities of Solutions of Second Order Quasilinear Equations, Longman, Harlow (1996).

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