Asymptotic behaviour of solutions of some nonlinear parabolic or elliptic equations

Kondratiev, V. А.; Верон, Лаурент

doi:10.3233/asy-1997-14202

Cited by 44 publications

(15 citation statements)

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“…Extinction properties for second order semilinear parabolic equations of diffusion-absorption type with nondegenerate (x, t)dependent absorptional potential was studied in [5,7,21,22]. V. Kondratiev, L. Veron [1] firstly initiated the study of EFT-property for second order equation (1.1) (m = 1) in the case of degenerate absorptional potential a(x):…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…V. Kondratiev, L. Veron [1] firstly initiated the study of EFT-property for second order equation (1.1) (m = 1) in the case of degenerate absorptional potential a(x):…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…It happens that occurence of mentioned property depends essentially on the structure of the set of degeneration and on the behaviour of potential a(x) in the neighbourhood of this set. They in [1] considered homogeneous Neumann problem for second order equation (1.1) (m = 1) and proved the following general sufficient condition for EFTproperty:…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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Extinction of solutions of semilinear higher order parabolic equations with degenerate absorption potential

Belaud

Shishkov²

2010

J. Evol. Equ.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

“…V. Kondratiev, L. Veron [1] firstly initiated the study of EFT-property for second order equation (1.1) (m = 1) in the case of degenerate absorptional potential a(x):…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: Introduction and Main Resultsmentioning

confidence: 99%

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Extinction of solutions of semilinear higher order parabolic equations with degenerate absorption potential

Belaud

Shishkov²

2010

J. Evol. Equ.

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“…for t 0, with constants C, δ > 0 independent of µ(α). This estimate with δ = 1/2 was supposed to be always true in [10], which we do not know; therefore in Theorem 4.3 (respectively 4.7) of this paper, inequality ∞ 0 µ −1 n < ∞ (respectively ∞ 0 µ −1/2 n < ∞) has to be replaced by ∞ 0 ln µ n /µ n < ∞ (respectively ∞ 0 ln µ n / √ µ n < ∞). However if any solution of (2.31) satisfies an estimate of the type The TCS-property admits a local version if we assume that the operator A reduces to a N × N bounded and measurable matrix A(x) = (a ij (x)) and r → f (x, r) is nondecreasing.…”

Section: The Time Compact Support Propertymentioning

confidence: 95%

“…Between those two extreme situations there exists a wide class of situations which were first explored by Kondratiev and Véron [10]. If n is an integer, they introduce the fundamental state of an associated Schrödinger operator µ n = inf Ω |∇ψ| 2 + 2 n b(x)ψ 2 dx: ψ ∈ W 1,2 (Ω), Ω ψ 2 dx = 1 , (1.4) and they proved that if…”

Section: 3)mentioning

confidence: 99%

Long-time vanishing properties of solutions of some semilinear parabolic equations

Belaud¹,

Helffer²,

Верон³

2001

Ann. Inst. H. Poincaré C Anal. Non Linéaire

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We study the long-time behavior of solutions of semilinear parabolic equations of the following type (PE) ∂ t u − ∇. A(x, t, u, ∇u) + f (x, u) = 0 where f (x, u) ≈ b(x)|u| q−1 u, b being a nonnegative bounded and measurable function and q a real number such that 0 q < 1. We give criteria which imply that any solution of the above equations vanishes in finite time and these criteria are associated to semi-classical limits of some Schrödinger operators. We also give a series of sufficient conditions on b(x) which imply that any supersolution with positive initial data does not to vanish identically for any positive t. © 2001 Éditions scientifiques et médicales Elsevier SAS RÉSUMÉ.-Nous étudions le comportement en temps grand de solutions d'équations paraboliques du type (PE) ∂ t u − ∇. A(x, t, u, ∇u) + f (x, u) = 0, où f (x, u) ≈ b(x)|u| q−1 u, b étant une fonction positive, bornée et mesurable, et q un nombre réel tel que 0 q < 1. Nous donnons des critères qui impliquent que toute solution des équations ci-dessus devient identiquement nulle en temps fini et ces critères sont associés à des problèmes de limite semiclassique d'opérateurs de Schrödinger. Nous donnons aussi une série de conditions suffisantes sur b(x) qui impliquent que toute sur-solution avec des données initiales positives ne devient jamais identiquement égale à zéro.

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Vladimir Aleksandrovich Kondrat'ev (A Tribute in Honor of His 70th Birthday)

Ильин¹,

Козлов²,

Садовничий³

et al. 2005

Diff Equat

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On July 2, 2005, Vladimir Aleksandrovich Kondrat'ev, Honored Professor of Moscow State University, Doctor of Physical and Mathematical Sciences), celebrated his seventieth birthday.Vladimir Aleksandrovich Kondrat'ev was born in Samara (earlier Kuibyshev), Russia, on July 2, 1935. His father was Professor of mechanics at Kuibyshev Industrial Institute, and his mother was a teacher of mathematics in a secondary school. In 1952, Kondrat'ev graduated from school no. 6 of Kuibyshev with excellence and entered the Mechanical-Mathematical Department of Moscow State University; in 1957, he graduated from the university. In 1959, a year before finishing postgraduate studies, he defended his Ph.D. thesis "On Zeros of Solutions of Linear Differential Equations of Order > 2" supervised by Professor S.A. Gal'pern, Doctor of Physical and Mathematical Sciences; in 1965, Kondrat'ev defended his D.Sc. thesis "Boundary Value Problems for Elliptic and Parabolic Equations with Singularities on the Boundary." Kondrat'ev's scientific interests were formed under the great influence of Academician I.G. Petrovskii. Since 1961, Vladimir Aleksandrovich Kondrat'ev has been working as Assistant Professor of the Chair of Differential Equations at the Mechanical-Mathematical Department of Moscow State University, and since 1970, he has been Professor of this chair.The main directions of Kondrat'ev's research are the following: the investigation of the oscillation property of solutions of ordinary differential equations; elliptic equations in domains with corners and conical points; qualitative and asymptotic properties of solutions of linear and nonlinear elliptic and parabolic equations and systems; spectral problems of differential operators; estimates for the minimum eigenvalue of differential operators; asymptotic behavior of solutions of ordinary differential equations with operator coefficients and qualitative properties of weak solutions of elliptic boundary value problems; analysis of mathematical problems of elasticity.Kondrat'ev's first scientific results were obtained during the study at the university and deal with the investigation of the oscillation property of solutions of linear ordinary differential equations. He justified a nonoscillation criterion for solutions of a second-order linear differential equation and 0012-2661/05/4107-0909 c 2005 Pleiades Publishing, Inc.

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Asymptotic behaviour of solutions of some nonlinear parabolic or elliptic equations

Cited by 44 publications

References 15 publications

Extinction of solutions of semilinear higher order parabolic equations with degenerate absorption potential

Extinction of solutions of semilinear higher order parabolic equations with degenerate absorption potential

Long-time vanishing properties of solutions of some semilinear parabolic equations

Vladimir Aleksandrovich Kondrat'ev (A Tribute in Honor of His 70th Birthday)

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