scholarly journals On the evolution equation with a dynamic Hardy-type potential

2021 ◽  
Author(s):  
Jann-Long Chern ◽  
Gyeongha Hwang ◽  
Jin Takahashi ◽  
Eiji Yanagida
Keyword(s):  
Integral Equation ◽  
Singular Point ◽  
Heat Equation ◽  
Existence And Uniqueness ◽  
Singular Potential ◽  
Initial Value ◽  
Uniqueness Of Solutions ◽  
Hardy Type ◽  
Radial Heat ◽  
Type Potential

Abstract Motivated by the celebrated paper of Baras and Goldstein (1984), we study the heat equation with a dynamic Hardy-type singular potential. In particular, we are interested in the case where the singular point moves in time. Under appropriate conditions on the potential and initial value, we show the existence, non-existence and uniqueness of solutions, and obtain a sharp lower and upper bound near the singular point. Proofs are given by using solutions of the radial heat equation, some precise estimates for an equivalent integral equation and the comparison principle.

scholarly journals To solving the fractionally loaded heat equation

2021 ◽  
Vol 101 (1) ◽  
pp. 65-77
Author(s):  
M.T. Kosmakova ◽  
◽  
S.A. Iskakov ◽  
L.Zh. Kasymova ◽  
◽  
...  
Keyword(s):  
Integral Equation ◽  
Boundary Value Problem ◽  
Heat Equation ◽  
Fractional Derivative ◽  
Existence And Uniqueness ◽  
Boundary Value ◽  
Continuous Functions ◽  
Uniqueness Of Solutions ◽  
Loaded Term ◽  
Differential Part

In this paper we consider a boundary value problem for a fractionally loaded heat equation in the class of continuous functions. Research methods are based on an approach to the study of boundary value problems, based on their reduction to integral equations. The problem is reduced to a Volterra integral equation of the second kind by inverting the differential part. We also carried out a study the limit cases for the fractional derivative order of the term with a load in the heat equation of the boundary value problem. It is shown that the existence and uniqueness of solutions to the integral equation depends on the order of the fractional derivative in the loaded term.

scholarly journals The Existence and Uniqueness of Solutions and Lyapunov-Type Inequality for CFR Fractional Differential Equations

2018 ◽  
Vol 2018 ◽  
pp. 1-7 ◽  
Author(s):  
Xia Wang ◽  
Run Xu
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Existence And Uniqueness ◽  
Type Inequality ◽  
Uniqueness Theorems ◽  
Initial Value ◽  
Uniqueness Of Solutions ◽  
Maximum Value ◽  
Fractional Differential ◽  
Lyapunov Type Inequality

In this paper, we research CFR fractional differential equations with the derivative of order 3<α<4. We prove existence and uniqueness theorems for CFR-type initial value problem. By Green’s function and its corresponding maximum value, we obtain the Lyapunov-type inequality of corresponding equations. As for application, we study the eigenvalue problem in the sense of CFR.

scholarly journals Theory of existence and uniqueness for the nonlinear Maxwell-Boltzmann equation I

1977 ◽  
Vol 16 (3) ◽  
pp. 379-414 ◽  
Author(s):  
Aleksander Glikson
Keyword(s):  
Boltzmann Equation ◽  
Existence And Uniqueness ◽  
Broad Class ◽  
Physical Space ◽  
Uniqueness Result ◽  
Cauchy Problems ◽  
Initial Value ◽  
Uniqueness Of Solutions ◽  
Boltzmann Gas ◽  
Definition Of

A review of the development of the theory of existence and uniqueness of solutions to initial-value problems for mostly reduced versions of the nonlinear Maxwell-Boltzmann equation with a cut-off of intermolecular interaction, precedes the formulation and discussion of a somewhat generalized initial-value problem for the full nonlinear Maxwell-Boltzmann equation, with or without a cut-off. This is followed by a derivation of a new existence-uniqueness result for a particular Cauchy problem for the full nonlinear Maxwell-Boltzmann equation with a cut-off, under the assumption that the monatomic Boltzmann gas in the unbounded physical space X is acted upon by a member of a broad class of external conservative forces with sufficiently well-behaved potentials, defined on X and bounded from below. The result represents a significant improvement of an earlier theorem by this author which was until now the strongest obtained for Cauchy problems for the full Maxwell-Boltzmann equation. The improvement is basically due to the introduction of equivalent norms in a Banach space, the definition of which is connected with an exponential function of the total energy of a free-streaming molecule.

THE INITIAL VALUE PROBLEM FOR THE DEEP BÉNARD CONVECTION EQUATIONS WITH DATA IN Lq

1996 ◽  
Vol 06 (02) ◽  
pp. 269-277 ◽  
Author(s):  
Z. CHARKI
Keyword(s):  
Fixed Point ◽  
Initial Value Problem ◽  
Existence And Uniqueness ◽  
Initial Value ◽  
Uniqueness Of Solutions ◽  
Bénard Convection ◽  
Benard Convection ◽  
Point Argument

A fixed point argument is used to prove the existence and uniqueness of solutions for the unsteady deep Bénard convection equations in [Formula: see text] for [Formula: see text].

Existence and uniqueness of solutions for singular integral equation

Positivity ◽  
2008 ◽  
Vol 12 (4) ◽  
pp. 725-732 ◽  
Author(s):  
Zhongwei Cao ◽  
Daqing Jiang ◽  
Chengjun Yuan ◽  
Donal O’Regan
Keyword(s):  
Integral Equation ◽  
Singular Integral Equation ◽  
Singular Integral ◽  
Existence And Uniqueness ◽  
Uniqueness Of Solutions

scholarly journals On Existence and Uniqueness of Solutions of a Nonlinear Integral Equation

2011 ◽  
Vol 2011 ◽  
pp. 1-7 ◽  
Author(s):  
M. Eshaghi Gordji ◽  
H. Baghani ◽  
O. Baghani
Keyword(s):  
Banach Space ◽  
Integral Equation ◽  
Fixed Point ◽  
Integral Operator ◽  
Existence And Uniqueness ◽  
Nonlinear Integral Equation ◽  
Uniqueness Of Solutions ◽  
Nonlinear Integral Operator

The purpose of this paper is to study the existence of fixed point for a nonlinear integral operator in the framework of Banach space . Later on, we give some examples of applications of this type of results.

scholarly journals Nonlocal problem with the integral condition for a loaded heate equation

2021 ◽  
pp. 47-56
Author(s):  
М.М. Сагдуллаева
Keyword(s):  
Integral Equation ◽  
Heat Equation ◽  
Regular Solution ◽  
Volterra Integral Equation ◽  
Existence And Uniqueness ◽  
Nonlocal Problem ◽  
Integral Condition ◽  
Local Problem ◽  
Non Local ◽  
Loaded Term

В работе рассмотрена нелокальная задача с интегральным условием для нагруженного уравнения теплопроводности, где нагруженное слагаемое представляет собой производную второго порядка от неизвестной функции в начале координат. Доказано существование и единственность регулярного решения. С помощью функции Грина и тепловых потенциалов доказанао существование регулярного решения исследуемой задачи. Доказательство основано на редукции поставленной задачи к интегральному уравнению Вольтерра второго рода со слабой особенностью. Из разрешимости полученных интегральных уравнений Вольтерра следует существование единственного решения поставленной задачи. In this paper, we consider a non-local problem with the integral condition for the loaded heat equation, where the loaded term is a derivative of the second order from an unknown function at the origin. The existence and uniqueness of a regular solution is proven. Using the Green’s functions and thermal potentials, the existence of a regular solution to this problem is proved. The proof is based on the reduction of the formulated problem to the second kind Volterra integral equation with a weak singularity. The solvability of the obtained Volterra integral equations implies the existence of a unique solution to the problem.

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