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 Insensitizing control for linear and semi-linear heat equations with partially unknown domain

2019 ◽  
Vol 25 ◽  
pp. 50 ◽  
Author(s):  
Pierre Lissy ◽  
Yannick Privat ◽  
Yacouba Simporé
Keyword(s):  
Heat Equation ◽  
Control Problem ◽  
Duality Theory ◽  
Unique Continuation ◽  
Dirichlet Boundary Conditions ◽  
Linear Case ◽  
Heat Equations ◽  
Geometrical Approach ◽  
Linear Heat Equation ◽  
Linear Heat

We consider a semi-linear heat equation with Dirichlet boundary conditions and globally Lipschitz nonlinearity, posed on a bounded domain of ℝN (N ∈ ℕ*), assumed to be an unknown perturbation of a reference domain. We are interested in an insensitizing control problem, which consists in finding a distributed control such that some functional of the state is insensitive at the first order to the perturbations of the domain. Our first result consists of an approximate insensitization property on the semi-linear heat equation. It rests upon a linearization procedure together with the use of an appropriate fixed point theorem. For the linear case, an appropriate duality theory is developed, so that the problem can be seen as a consequence of well-known unique continuation theorems. Our second result is specific to the linear case. We show a property of exact insensitization for some families of deformation given by one or two parameters. Due to the nonlinearity of the intrinsic control problem, no duality theory is available, so that our proof relies on a geometrical approach and direct computations.

Global blowup controllability of heat equation with feedback control

2018 ◽  
Vol 20 (05) ◽  
pp. 1750062 ◽  
Author(s):  
Ping Lin
Keyword(s):  
Feedback Control ◽  
Initial Data ◽  
Heat Equation ◽  
Heat Equations ◽  
Controllability Problem ◽  
Space Domain ◽  
Global Controllability ◽  
Linear Heat ◽  
Heat System

This paper concerns a global controllability problem for heat equations. In the absence of control, the solution to the linear heat system globally exists. While for each initial data, we can find a feedback control acting on an internal subset of the space domain such that the corresponding solution to the system blows up at given time.

scholarly journals On Positive Solutions of the Heat Equation

1967 ◽  
Vol 30 ◽  
pp. 203-207 ◽  
Author(s):  
Masasumi Kato
Keyword(s):  
Cauchy Problem ◽  
Euclidean Space ◽  
Heat Equation ◽  
Uniqueness Theorem ◽  
Positive Solutions ◽  
Dimensional Euclidean Space ◽  
The Cauchy Problem ◽  
The Uniqueness Theorem

Consider the positive and twice continuously differentiable solutions u of the heat equationin an open t-strip Ω = Rn×(0,T) for some T>0, where Rn is the n-dimensional Euclidean space.In this note, we prove a theorem of Fatou type on u and, as its application, the uniqueness theorem for the Cauchy problem of ( 1 ).

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>

Conserved Quantities for the General Linear Heat Equation

2016 ◽  
pp. 4437-4439
Author(s):  
Adil Jhangeer ◽  
Fahad Al-Mufadi
Keyword(s):  
Evolution Equation ◽  
Heat Equation ◽  
Conservation Laws ◽  
Exact Solutions ◽  
Lagrangian Approach ◽  
General Linear ◽  
Conserved Quantities ◽  
Linear Heat ◽  
The Family ◽  
Partial Lagrangian

In this paper, conserved quantities are computed for a class of evolution equation by using the partial Noether approach [2]. The partial Lagrangian approach is applied to the considered equation, infinite many conservation laws are obtained depending on the coefficients of equation for each n. These results give potential systems for the family of considered equation, which are further helpful to compute the exact solutions.

Time and space meshes adaptivity for the resolution of a semi-linear heat equation

2021 ◽  
pp. 1-10
Author(s):  
Nejmeddine Chorfi
Keyword(s):  
Parabolic Equation ◽  
Heat Equation ◽  
Order Of Convergence ◽  
Linear Parabolic Equation ◽  
Spectral Elements ◽  
Euler Scheme ◽  
Time Step ◽  
Implicit Euler Scheme ◽  
Linear Heat ◽  
Error Indicators

The aim of this work is to highlight that the adaptivity of the time step when combined with the adaptivity of the spectral mesh is optimal for a semi-linear parabolic equation discretized by an implicit Euler scheme in time and spectral elements method in space. The numerical results confirm the optimality of the order of convergence. The later is similar to the order of the error indicators.

Sign in / Sign up

Export Citation Format

Share Document