Asymptotic behavior of elliptic nonlocal equations set in cylinders

2014 ◽  
Vol 89 (1-2) ◽  
pp. 21-35 ◽  
Author(s):  
Karen Yeressian
Keyword(s):  
Asymptotic Behavior ◽  
Nonlocal Equations

scholarly journals On the asymptotic analysis of problems involving fractional Laplacian in cylindrical domains tending to infinity

2016 ◽  
Vol 19 (05) ◽  
pp. 1650035 ◽  
Author(s):  
Indranil Chowdhury ◽  
Prosenjit Roy
Keyword(s):  
Asymptotic Behavior ◽  
Asymptotic Analysis ◽  
Fractional Laplacian ◽  
Elliptic Problems ◽  
Weighted Estimate ◽  
Dirichlet Problems ◽  
Nonlocal Equations ◽  
Cylindrical Domains ◽  
Non Local ◽  
Second Order Elliptic Problems

The paper is an attempt to investigate the issues of asymptotic analysis for problems involving fractional Laplacian where the domains tend to become unbounded in one-direction. Motivated from the pioneering work on second-order elliptic problems by Chipot and Rougirel in [On the asymptotic behaviour of the solution of elliptic problems in cylindrical domains becoming unbounded, Commun. Contemp. Math. 4(1) (2002) 15–44], where the force functions are considered on the cross-section of domains, we prove the non-local counterpart of their result.Recently in [Asymptotic behavior of elliptic nonlocal equations set in cylinders, Asymptot. Anal. 89(1–2) (2014) 21–35] Yeressian established a weighted estimate for solutions of non-local Dirichlet problems which exhibit the asymptotic behavior. The case when [Formula: see text] was also treated as an example to show how the weighted estimate might be used to achieve the asymptotic behavior. In this paper, we extend this result to each order between [Formula: see text] and [Formula: see text].

scholarly journals Profile of solutions for nonlocal equations with critical and supercritical nonlinearities

2019 ◽  
Vol 21 (01) ◽  
pp. 1750099 ◽  
Author(s):  
Mousomi Bhakta ◽  
Debangana Mukherjee ◽  
Sanjiban Santra
Keyword(s):  
Asymptotic Behavior ◽  
Positive Solution ◽  
Interior Point ◽  
Fractional Laplacian ◽  
Smooth Boundary ◽  
Symmetric Domain ◽  
Uniqueness Of Solution ◽  
Local Uniqueness ◽  
Nonlocal Equations ◽  
Shaped Domain

We study the fractional Laplacian problem [Formula: see text] where [Formula: see text], [Formula: see text] and [Formula: see text] is a parameter. Here, [Formula: see text] is a bounded star-shaped domain with smooth boundary and [Formula: see text]. We establish existence of a variational positive solution [Formula: see text] and characterize the asymptotic behavior of [Formula: see text] as [Formula: see text]. When [Formula: see text], we describe how the solution [Formula: see text] blows up at an interior point of [Formula: see text]. Furthermore, we prove the local uniqueness of solution of the above problem when [Formula: see text] is a convex symmetric domain of [Formula: see text] with [Formula: see text] and [Formula: see text].

scholarly journals Parabolic equations with natural growth approximated by nonlocal equations

2020 ◽  
Vol 23 (01) ◽  
pp. 1950088
Author(s):  
Tommaso Leonori ◽  
Alexis Molino ◽  
Sergio Segura de León
Keyword(s):  
Boundary Condition ◽  
Asymptotic Behavior ◽  
Comparison Principle ◽  
Dirichlet Boundary Condition ◽  
Parabolic Equations ◽  
Dirichlet Boundary ◽  
Natural Growth ◽  
Nonlocal Problems ◽  
Nonlocal Equations ◽  
Bounded Smooth

In this paper, we study several aspects related with solutions of nonlocal problems whose prototype is [Formula: see text] where we take, as the most important instance, [Formula: see text] with [Formula: see text] as well as [Formula: see text], [Formula: see text] is a smooth symmetric function with compact support and [Formula: see text] is either a bounded smooth subset of [Formula: see text], with nonlocal Dirichlet boundary condition, or [Formula: see text] itself. The results deal with existence, uniqueness, comparison principle and asymptotic behavior. Moreover, we prove that if the kernel is rescaled in a suitable way, the unique solution of the above problem converges to a solution of the deterministic Kardar–Parisi–Zhang equation.

Sign in / Sign up

Export Citation Format

Share Document