scholarly journals A matrix Harnack inequality for semilinear heat equations

2022 ◽  
Vol 5 (1) ◽  
pp. 1-15
Author(s):  
Giacomo Ascione ◽  
◽  
Daniele Castorina ◽  
Giovanni Catino ◽  
Carlo Mantegazza ◽  
...  
Keyword(s):  
Heat Equation ◽  
Positive Solutions ◽  
Harnack Inequality ◽  
Riemannian Manifolds ◽  
Ricci Tensor ◽  
Heat Equations ◽  
Standard Heat ◽  
Matrix Version ◽  
Sectional Curvatures ◽  
Parallel Ricci Tensor

<abstract><p>We derive a matrix version of Li &amp; Yau–type estimates for positive solutions of semilinear heat equations on Riemannian manifolds with nonnegative sectional curvatures and parallel Ricci tensor, similarly to what R. Hamilton did in <sup>[<xref ref-type="bibr" rid="b5">5</xref>]</sup> for the standard heat equation. We then apply these estimates to obtain some Harnack–type inequalities, which give local bounds on the solutions in terms of the geometric quantities involved.</p></abstract>

scholarly journals Generating monotone quantities for the heat equation

2019 ◽  
Vol 2019 (756) ◽  
pp. 37-63 ◽  
Author(s):  
Jonathan Bennett ◽  
Neal Bez
Keyword(s):  
Euclidean Space ◽  
Heat Equation ◽  
Positive Solutions ◽  
Algebraic Closure ◽  
Heat Equations ◽  
Differential Inequalities ◽  
Closure Properties ◽  
Rich Variety ◽  
Linear Heat ◽  
Convexity Properties

AbstractThe purpose of this article is to expose and further develop a simple yet surprisingly far-reaching framework for generating monotone quantities for positive solutions to linear heat equations in euclidean space. This framework is intimately connected to the existence of a rich variety of algebraic closure properties of families of sub/super-solutions, and more generally solutions of systems of differential inequalities capturing log-convexity properties such as the Li–Yau gradient estimate. Various applications are discussed, including connections with the general Brascamp–Lieb inequality and the Ornstein–Uhlenbeck semigroup.

scholarly journals Brownian Motion and Harmonic Analysis on Sierpinski Carpets

1999 ◽  
Vol 51 (4) ◽  
pp. 673-744 ◽  
Author(s):  
Martin T. Barlow ◽  
Richard F. Bass
Keyword(s):  
Brownian Motion ◽  
Heat Equation ◽  
Harmonic Analysis ◽  
Heat Kernel ◽  
Lower Bounds ◽  
Harnack Inequality ◽  
Harmonic Functions ◽  
Upper And Lower Bounds ◽  
Isotropic Diffusion ◽  
Sierpinski Carpets

AbstractWe consider a class of fractal subsets of d formed in a manner analogous to the construction of the Sierpinski carpet. We prove a uniform Harnack inequality for positive harmonic functions; study the heat equation, and obtain upper and lower bounds on the heat kernel which are, up to constants, the best possible; construct a locally isotropic diffusion X and determine its basic properties; and extend some classical Sobolev and Poincaré inequalities to this setting.

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