On a non-local problem for irregular equations

2000 ◽  
Vol 191 (11) ◽  
pp. 1607-1633
Author(s):  
V V Kornienko
Keyword(s):  
Local Problem ◽  
Non Local

scholarly journals Non-local problem with integral conditions for the three-dimensional high order hyperbolic equations

2020 ◽  
pp. 51-62
Author(s):  
Ш.Ш. Юсубов
Keyword(s):  
Integral Equation ◽  
Hyperbolic Equation ◽  
Hyperbolic Equations ◽  
Three Dimensional ◽  
High Order ◽  
Mixed Derivative ◽  
Local Problem ◽  
Integral Conditions ◽  
Order Hyperbolic Equation ◽  
Non Local

В работе для трехмерного гиперболического уравнения высокого порядка с доминирующей смешанной производной исследуется разрешимость нелокальной задачи с интегральными условиями. Поставленная задача сводится к интегральному уравнению и с помощью априорных оценок доказывается существование единственного решения. In the work the solvability of the non-local problem with integral conditions is investigated for the three-dimensional high order hyperbolic equation with dominated mixed derivative. The problem is reduced to the integral equation and existence of the solution is proved by the help of aprior estimations.

Multiplicity results for a non-local problem with concave and convex nonlinearities

2019 ◽  
Vol 182 ◽  
pp. 263-279 ◽  
Author(s):  
Najmeh Kouhestani ◽  
Hakimeh Mahyar ◽  
Abbas Moameni
Keyword(s):  
Local Problem ◽  
Multiplicity Results ◽  
Non Local ◽  
Concave And Convex Nonlinearities

scholarly journals Non-local to local transition for ground states of fractional Schrödinger equations on $$\mathbb {R}^N$$

2020 ◽  
Vol 22 (3) ◽  
Author(s):  
Bartosz Bieganowski ◽  
Simone Secchi
Keyword(s):  
Ground State ◽  
Ground States ◽  
Ground State Solution ◽  
State Solution ◽  
Local Problem ◽  
Schrödinger Equations ◽  
Ground State Solutions ◽  
Fractional Schrödinger Equations ◽  
Non Local ◽  
Local Transition

Abstract We consider the nonlinear fractional problem $$\begin{aligned} (-\Delta )^{s} u + V(x) u = f(x,u)&\quad \hbox {in } \mathbb {R}^N \end{aligned}$$ ( - Δ ) s u + V ( x ) u = f ( x , u ) in R N We show that ground state solutions converge (along a subsequence) in $$L^2_{\mathrm {loc}} (\mathbb {R}^N)$$ L loc 2 ( R N ) , under suitable conditions on f and V, to a ground state solution of the local problem as $$s \rightarrow 1^-$$ s → 1 - .

scholarly journals Unique solvability of a non-local problem for mixed-type equation with fractional derivative

2016 ◽  
Vol 40 (8) ◽  
pp. 2994-2999 ◽  
Author(s):  
Erkinjon T. Karimov ◽  
Abdumauvlen S. Berdyshev ◽  
Nilufar A. Rakhmatullaeva
Keyword(s):  
Fractional Derivative ◽  
Unique Solvability ◽  
Mixed Type ◽  
Type Equation ◽  
Local Problem ◽  
Mixed Type Equation ◽  
Non Local

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.

scholarly journals A gluing approach for the fractional Yamabe problem with isolated singularities

2020 ◽  
Vol 2020 (763) ◽  
pp. 25-78 ◽  
Author(s):  
Weiwei Ao ◽  
Azahara DelaTorre ◽  
María del Mar González ◽  
Juncheng Wei
Keyword(s):  
Approximate Solution ◽  
Reduction Method ◽  
Type System ◽  
Singular Solutions ◽  
Yamabe Problem ◽  
Local Problem ◽  
Infinite Dimensional ◽  
Isolated Points ◽  
Non Local ◽  
First Time

AbstractWe construct solutions for the fractional Yamabe problem that are singular at a prescribed number of isolated points. This seems to be the first time that a gluing method is successfully applied to a non-local problem in order to construct singular solutions. There are two main steps in the proof: to construct an approximate solution by gluing half bubble towers at each singular point, and then an infinite-dimensional Lyapunov–Schmidt reduction method, that reduces the problem to an (infinite-dimensional) Toda-type system. The main technical part is the estimate of the interactions between different bubbles in the bubble towers.

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