scholarly journals Well-posedness of the linear heat equation with a second order memory term

2020 ◽  
Vol 34 ◽  
pp. 03011
Author(s):  
Constantin Niţă ◽  
Laurenţiu Emanuel Temereancă
Keyword(s):  
Initial Data ◽  
Heat Equation ◽  
Unique Solution ◽  
Source Term ◽  
Memory Term ◽  
Well Posedness ◽  
Dimensional Torus ◽  
One Dimensional ◽  
Linear Heat ◽  
The One

In this article we prove that the heat equation with a memory term on the one-dimensional torus has a unique solution and we study the smoothness properties of this solution. These properties are related with some smoothness assumptions imposed to the initial data of the problem and to the source term.

scholarly journals An optimal time control problem for the one-dimensional, linear heat equation, in the presence of a scaling parameter

2017 ◽  
Vol 51 (4) ◽  
pp. 1289-1299 ◽  
Author(s):  
Karim Benalia ◽  
Claire David ◽  
Brahim Oukacha
Keyword(s):  
Heat Equation ◽  
Optimal Time ◽  
One Dimensional ◽  
Scaling Parameter ◽  
Linear Heat Equation ◽  
Linear Heat ◽  
Time Control ◽  
Time Problem ◽  
Type Property ◽  
The One

In this paper, we study the optimal time problem for the one-dimensional, linear heat equation, in the presence of a scaling parameter. To begin with, we build an exact solution. The dependence of this solution as regards the scaling parameter naturally opens the way to study the existence and uniqueness of an optimal time control. If, moreover, one assumes the L∞ − null controllability, it enables to establish a bang-bang type property.

scholarly journals The one-dimensional heat equation in the Alexiewicz norm

2015 ◽  
Vol 0 (0) ◽  
Author(s):  
Erik Talvila
Keyword(s):  
Banach Space ◽  
Initial Data ◽  
Heat Equation ◽  
Real Line ◽  
Distributional Derivative ◽  
One Dimensional ◽  
The Real ◽  
Space Of Distributions ◽  
The One

AbstractA distribution on the real line has a continuous primitive integral if it is the distributional derivative of a function that is continuous on the extended real line. The space of distributions integrable in this sense is a Banach space that includes all functions integrable in the Lebesgue and Henstock–Kurzweil senses. The one-dimensional heat equation is considered with initial data that is integrable in the sense of the continuous primitive integral. Let Θ

Global well-posedness of the Cauchy problem for the equations of a one-dimensional viscous heat-conducting gas with Lebesgue initial data

2004 ◽  
Vol 134 (5) ◽  
pp. 939-960 ◽  
Author(s):  
Song Jiang ◽  
Alexander Zlotnik
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Heat Conducting ◽  
Well Posedness ◽  
Global Weak Solutions ◽  
One Dimensional ◽  
Lipschitz Continuous ◽  
Viscous Heat ◽  
The Cauchy Problem ◽  
The One

We study the Cauchy problem for the one-dimensional equations of a viscous heat-conducting gas in the Lagrangian mass coordinates with the initial data in the Lebesgue spaces. We prove the existence, the uniqueness and the Lipschitz continuous dependence on the initial data of global weak solutions.

scholarly journals On the Reachable Set for the One-Dimensional Heat Equation

2018 ◽  
Vol 56 (3) ◽  
pp. 1692-1715 ◽  
Author(s):  
Jérémi Dardé ◽  
Sylvain Ervedoza
Keyword(s):  
Heat Equation ◽  
Reachable Set ◽  
One Dimensional ◽  
The One
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