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Global positivity estimates and Harnack inequalities for the fast diffusion equation
 J. Funct. Anal
"... We investigate local and global properties of positive solutions to the fast diffusion equation ut = ∆u m in the range (d − 2)+/d < m < 1, corresponding to general nonnegative initial data. For the Cauchy problem posed in the whole Euclidean space R d we prove sharp Local Positivity Estimates ..."
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Cited by 14 (10 self)
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We investigate local and global properties of positive solutions to the fast diffusion equation ut = ∆u m in the range (d − 2)+/d < m < 1, corresponding to general nonnegative initial data. For the Cauchy problem posed in the whole Euclidean space R d we prove sharp Local Positivity Estimates (Weak Harnack Inequalities) and Elliptic Harnack inequalities; we use them to derive sharp Global Positivity Estimates and a Global Harnack Principle. For the mixed initial and boundary value problem posed in a bounded domain of R d with homogeneous Dirichlet condition, we prove Weak and Elliptic Harnack Inequalities. Our work shows that these fast diffusion flows have regularity properties comparable and in some senses better than the linear heat flow.
On Fundamental Solutions of Generalized Schrödinger Operators
"... . We consider the generalized Schrodinger operator \Gamma\Delta + where is a nonnegative Radon measure in R n , n 3. Assuming that satisfies certain scaleinvariant Kato condition and doubling condition, we establish the following bounds for the fundamental solution of \Gamma\Delta + in R n ..."
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Cited by 6 (0 self)
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. We consider the generalized Schrodinger operator \Gamma\Delta + where is a nonnegative Radon measure in R n , n 3. Assuming that satisfies certain scaleinvariant Kato condition and doubling condition, we establish the following bounds for the fundamental solution of \Gamma\Delta + in R n : c e \Gamma" 2 d(x;y;) jx \Gamma yj n\Gamma2 \Gamma (x; y) C e \Gamma" 1 d(x;y;) jx \Gamma yj n\Gamma2 where d(x; y; ) is the distance function for the modified Agmon metric m(x; )dx 2 associated with . We also study the boundedness of the corresponding Riesz transforms r(\Gamma\Delta + ) \Gamma1=2 on L p (R n ; dx). Keywords. Schrodinger Operators, Fundamental Solutions, Riesz Transforms. Introduction Consider the generalized Schrodinger operator (0.1) \Gamma\Delta + in R n ; n 3 where is a nonnegative Radon measure on R n . The main purpose of this paper is to establish optimal upper and lower bounds for the fundamental solution of \Gamma\Delta + under suita...
The AllegrettoPiepenbrink Theorem for Strongly Local Dirichlet Forms
 DOCUMENTA MATH.
, 2009
"... The existence of positive weak solutions is related to spectral information on the corresponding partial differential operator. ..."
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Cited by 6 (5 self)
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The existence of positive weak solutions is related to spectral information on the corresponding partial differential operator.
Generalized eigenfunctions and spectral theory for strongly local Dirichlet forms
, 2009
"... We present an introduction to the framework of strongly local Dirichlet forms and discuss connections between the existence of certain generalized eigenfunctions and spectral properties within this framework. The range of applications is illustrated by a list of examples. ..."
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Cited by 4 (4 self)
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We present an introduction to the framework of strongly local Dirichlet forms and discuss connections between the existence of certain generalized eigenfunctions and spectral properties within this framework. The range of applications is illustrated by a list of examples.
Global asymptotic behavior of solutions of semilinear parabolic equation
 abdeljabbar.ghanmi@lamsin.rnu.tn Abdeljabbar Ghanmi and Faten Toumi Faculté des Sciences de Tunis, Département de Mathématiques, Campus Universitaire, 2092
"... Abstract. We study the large time behavior of solutions for the semilinear parabolic equation ∆u+V up−ut = 0. Under a general and natural condition on V = V (x) and the initial value u0, we show that global positive solutions of the parabolic equation converge pointwise to positive solutions of the ..."
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Cited by 3 (0 self)
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Abstract. We study the large time behavior of solutions for the semilinear parabolic equation ∆u+V up−ut = 0. Under a general and natural condition on V = V (x) and the initial value u0, we show that global positive solutions of the parabolic equation converge pointwise to positive solutions of the corresponding elliptic equation. As a corollary of this, we recapture the global existence results on semilinear elliptic equations obtained by Kenig and Ni and by F.H. Lin and Z. Zhao. Our method depends on newly found global bounds for fundamental solutions of certain linear parabolic equations. 1.
A Green function and regularity results for an ultraparabolic equation with a singular potential
 Adv. in Diff. Eq
"... Abstract. We prove a Harnack inequality for the positive solutions of a Schrödinger type equation L0 u + V u = 0, where L0 is an operator satisfying the Hörmander’s condition and V belongs to a class of functions of StummelKato type. We also obtain the existence of a Green function and an uniquenes ..."
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Cited by 2 (2 self)
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Abstract. We prove a Harnack inequality for the positive solutions of a Schrödinger type equation L0 u + V u = 0, where L0 is an operator satisfying the Hörmander’s condition and V belongs to a class of functions of StummelKato type. We also obtain the existence of a Green function and an uniqueness result for the CauchyDirichlet problem. 1.
Long time behavior of Riemannian mean curvature flow of graphs
 J. Math. Anal. Appl
"... curvature flow of graphs ..."
UNIFORM INTEGRABILITY OF APPROXIMATE GREEN FUNCTIONS OF SOME DEGENERATE ELLIPTIC OPERATORS
, 2001
"... We prove the uniformintegrability of the approximate Green functions of some degenerate elliptic operators in divergence formwith lower order termcoefficients satisfying a Kato type condition. Some further properties of the approximate Green functions of such operators are also established. 1. Intro ..."
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We prove the uniformintegrability of the approximate Green functions of some degenerate elliptic operators in divergence formwith lower order termcoefficients satisfying a Kato type condition. Some further properties of the approximate Green functions of such operators are also established. 1. Introduction. In this paper, we study the approximate Green functions of certain degenerate elliptic operators L on balls in Rn,n>2, when L has the divergence form
A FEFFERMANPOINCARE ́ TYPE INEQUALITY FOR CARNOTCARATHÉODORY VECTOR FIELDS
, 2002
"... Dedicated to Professor Michele Frasca on the occasion of his sixtieth birthday Abstract. In this note we prove a FeffermanPoincare ́ type inequality in spaces with metric induced by CarnotCarathéodory vector fields. 1. ..."
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Dedicated to Professor Michele Frasca on the occasion of his sixtieth birthday Abstract. In this note we prove a FeffermanPoincare ́ type inequality in spaces with metric induced by CarnotCarathéodory vector fields. 1.
Research Article Harnack Inequality for the Schrödinger Problem Relative to Strongly Local Riemannian pHomogeneous Forms with a Potential in the Kato Class
"... We define a notion of Kato class of measures relative to a Riemannian strongly local phomogeneous Dirichlet form and we prove a Harnack inequality (on balls that are small enough) for the positive solutions to a Schrödingertype problem relative to the form with a potential in the Kato class. Copy ..."
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We define a notion of Kato class of measures relative to a Riemannian strongly local phomogeneous Dirichlet form and we prove a Harnack inequality (on balls that are small enough) for the positive solutions to a Schrödingertype problem relative to the form with a potential in the Kato class. Copyright © 2007 M. Biroli and S. Marchi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1.