Characterization of Spaces of Bessel Potentials Related to the Heat Equation

2010 ◽  
pp. 143-155
Author(s):  
B. Frank Jones
Keyword(s):  
Heat Equation ◽  
Bessel Potentials

scholarly journals Flett Potential Spaces and a Weighted Wavelet-Like Transform

2020 ◽  
Vol 10 (3) ◽  
pp. 112-122
Author(s):  
Sinem Sezer Evcan ◽  
◽  
Sevda Barut
Keyword(s):  
Finite Differences ◽  
Bessel Potentials ◽  
Poisson Semigroup ◽  
Classical Potential

In this study, the Flett potential spaces are defined and a characterization of these potential spaces is given. Most of the known characterizations of classical potential spaces such as Riesz, Bessel potentials spaces and their generalizations are given in terms of finite differences. Here, by taking wavelet measure instead of finite differences, a weighted wavelet-like transform associated with Poisson semigroup is defined. And, by making use of this weighted wavelet-like transform, a new “truncated" integrals are defined, then using these integrals a characterization of the Flett potential spaces is given.

scholarly journals A CHARACTERIZATION OF SOBOLEV SPACES BY SOLUTIONS OF HEAT EQUATION AND A STABILITY PROBLEM FOR A FUNCTIONAL EQUATION

2008 ◽  
Vol 23 (3) ◽  
pp. 401-411
Author(s):  
Yun-Sung Chung ◽  
Young-Su Lee ◽  
Deok-Yong Kwon ◽  
Soon-Yeong Chung
Keyword(s):  
Functional Equation ◽  
Heat Equation ◽  
Sobolev Spaces ◽  
Stability Problem

scholarly journals Characterization of Riesz and Bessel potentials on variable Lebesgue spaces

2006 ◽  
Vol 4 (2) ◽  
pp. 113-144 ◽  
Author(s):  
Alexandre Almeida ◽  
Stefan Samko
Keyword(s):  
Sobolev Spaces ◽  
Variable Exponent ◽  
Regularity Conditions ◽  
Bessel Potential ◽  
Lebesgue Spaces ◽  
Hypersingular Integrals ◽  
Variable Lebesgue Spaces ◽  
Bessel Potentials ◽  
Bessel Potential Spaces

Riesz and Bessel potential spaces are studied within the framework of the Lebesgue spaces with variable exponent. It is shown that the spaces of these potentials can be characterized in terms of convergence of hypersingular integrals, if one assumes that the exponent satisfies natural regularity conditions. As a consequence of this characterization, we describe a relation between the spaces of Riesz or Bessel potentials and the variable Sobolev spaces.

scholarly journals The determination of caloric morphisms on Euclidean domains

2000 ◽  
Vol 158 ◽  
pp. 133-166 ◽  
Author(s):  
Katsunori Shimomura
Keyword(s):  
Heat Equation ◽  
Positive Function ◽  
Smooth Mapping ◽  
Direct Sum ◽  
Smooth Positive Function ◽  
Euclidean Domains

AbstractLetDbe a domain in ℝm+1andEbe a domain in ℝn+1. A pair of a smooth mappingf:D → Eand a smooth positive function ϕ onDis called a caloric morphism if ϕ˙uofis a solution of the heat equation inDwheneveruis a solution of the heat equation inE. We give the characterization of caloric morphisms, and then give the determination of caloric morphisms. In the case ofm < n, there are no caloric morphisms. In the case ofm = n, caloric morphisms are generated by the dilation, the rotation, the translation and the Appell transformation. In the case ofm > n, under some assumption onf, every caloric morphism is obtained by composing a projection with a direct sum of caloric morphisms of ℝn+1.

scholarly journals A concrete realization of the slow-fast alternative for a semilinear heat equation with homogeneous Neumann boundary conditions

2018 ◽  
Vol 7 (3) ◽  
pp. 375-384 ◽  
Author(s):  
Marina Ghisi ◽  
Massimo Gobbino ◽  
Alain Haraux
Keyword(s):  
Boundary Conditions ◽  
Heat Equation ◽  
Linear Part ◽  
Neumann Boundary ◽  
Neumann Boundary Conditions ◽  
Asymptotic Behavior Of Solutions ◽  
Semilinear Heat Equation ◽  
Codimension One ◽  
Concrete Realization

AbstractWe investigate the asymptotic behavior of solutions to a semilinear heat equation with homogeneous Neumann boundary conditions. It was recently shown that the nontrivial kernel of the linear part leads to the coexistence of fast solutions decaying to 0 exponentially (as time goes to infinity), and slow solutions decaying to 0 as negative powers of t. Here we provide a characterization of slow/fast solutions in terms of their sign, and we show that the set of initial data giving rise to fast solutions is a graph of codimension one in the phase space.

scholarly journals Ill-conditioning versus ill-posedness for the boundary controllability of the heat equation

2015 ◽  
Vol 23 (4) ◽  
Author(s):  
Faker Ben Belgacem ◽  
Sidi Mahmoud Kaber
Keyword(s):  
Heat Equation ◽  
Exact Controllability ◽  
Adjoint Equations ◽  
Boundary Controllability ◽  
The Troubles ◽  
Hilbert Uniqueness Method ◽  
Ill Posed ◽  
Uniqueness Method ◽  
Ill Conditioning

AbstractIll-posedness and/or ill-conditioning are features users have to deal with appropriately in the controllability of diffusion problems for secure and reliable outputs. We investigate those issues in the case of a boundary Dirichlet control, in an attempt to underline the origin of the troubles arising in the numerical computations and to shed some light on the difficulties to obtain good quality simulations. The exact-controllability is severely ill-posed while, in spite of its well-posedness, the null-controllability turns out to be very badly ill-conditioned. Theoretical and numerical results are stated on the heat equation in one dimension to illustrate the specific instabilities of each problem. The main tools used here are first a characterization of the subspace where the HUM (Hilbert Uniqueness Method) control lies and the study of the spectrum of some structured matrices, of Pick and Löwner type, obtained from the Fourier calculations on the state and adjoint equations.

Series Expansions for Dual Laguerre Temperatures

1972 ◽  
Vol 24 (6) ◽  
pp. 1145-1153 ◽  
Author(s):  
Deborah Tepper Haimo
Keyword(s):  
Heat Equation ◽  
Double Series ◽  
Series Expansions ◽  
Differential Heat ◽  
Generalized Heat Equation

In a recent paper [2], the author, with F. M. Cholewinski, derived criteria for the series expansions of solutions u(x, t) of the Laguerre differential heat equation xuxx + (α + 1 - x)ux = ut in terms of the Laguerre heat polynomials and of their temperature transforms. Our present goal is the characterization of those solutions which are representable in a Maclaurin double series in xe-t and in 1 — e-t Some of the results are analogous to those derived by D. V. Widder in [4] for the classical heat equation and by the author in [1] for the generalized heat equation.

Characterization of Fourier hyperfunctions by solutions of the Hermite heat equation

2007 ◽  
Vol 18 (7) ◽  
pp. 471-480 ◽  
Author(s):  
Bishnu P. Dhungana ◽  
Soon-Yeong Chung ◽  
Dohan Kim
Keyword(s):  
Heat Equation ◽  
Hermite Heat Equation ◽  
Fourier Hyperfunctions
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