scholarly journals A general blow-up result for a degenerate hyperbolic inequality in an exterior domain

2021 ◽  
Author(s):  
Mohamed Jleli ◽  
Mokhtar Kirane ◽  
Bessem Samet
Keyword(s):  
Boundary Conditions ◽  
Critical Exponent ◽  
Exterior Domain ◽  
Critical Case ◽  
Blow Up ◽  
Unified Approach ◽  
Dirichlet Type ◽  
Open Questions ◽  
Neumann Type ◽  
Type Boundary

In this paper, we consider a degenerate hyperbolic inequality in an exterior domain under three types of boundary conditions: Dirichlet-type, Neumann-type, and Robin-type boundary conditions. Using a unified approach, we show that all the considered problems have the same Fujita critical exponent. Moreover, we answer some open questions from the literature regarding the critical case.

scholarly journals An asymptotically compatible approach for Neumann-type boundary condition on nonlocal problems

2020 ◽  
Vol 54 (4) ◽  
pp. 1373-1413 ◽  
Author(s):  
Huaiqian You ◽  
XinYang Lu ◽  
Nathaniel Task ◽  
Yue Yu
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Nonlocal Boundary ◽  
Nonlocal Diffusion ◽  
Order Convergence ◽  
Flux Boundary Condition ◽  
Nonlocal Interactions ◽  
Local Case ◽  
Neumann Type ◽  
Type Boundary

In this paper we consider 2D nonlocal diffusion models with a finite nonlocal horizon parameter δ characterizing the range of nonlocal interactions, and consider the treatment of Neumann-like boundary conditions that have proven challenging for discretizations of nonlocal models. We propose a new generalization of classical local Neumann conditions by converting the local flux to a correction term in the nonlocal model, which provides an estimate for the nonlocal interactions of each point with points outside the domain. While existing 2D nonlocal flux boundary conditions have been shown to exhibit at most first order convergence to the local counter part as δ → 0, the proposed Neumann-type boundary formulation recovers the local case as O(δ2) in the L∞ (Ω) norm, which is optimal considering the O(δ2) convergence of the nonlocal equation to its local limit away from the boundary. We analyze the application of this new boundary treatment to the nonlocal diffusion problem, and present conditions under which the solution of the nonlocal boundary value problem converges to the solution of the corresponding local Neumann problem as the horizon is reduced. To demonstrate the applicability of this nonlocal flux boundary condition to more complicated scenarios, we extend the approach to less regular domains, numerically verifying that we preserve second-order convergence for non-convex domains with corners. Based on the new formulation for nonlocal boundary condition, we develop an asymptotically compatible meshfree discretization, obtaining a solution to the nonlocal diffusion equation with mixed boundary conditions that converges with O(δ2) convergence.

NUMERICAL APPROXIMATION OF VISCOSITY SOLUTIONS OF FIRST-ORDER HAMILTON-JACOBI EQUATIONS WITH NEUMANN TYPE BOUNDARY CONDITIONS

1992 ◽  
Vol 02 (03) ◽  
pp. 357-374 ◽  
Author(s):  
ELISABETH ROUY
Keyword(s):  
Boundary Conditions ◽  
Viscosity Solutions ◽  
Numerical Approximation ◽  
Time Discretization ◽  
Finite Difference Schemes ◽  
First Order ◽  
The Cauchy Problem ◽  
Neumann Type ◽  
Type Boundary ◽  
Jacobi Equations

We present a general result concerning numerical approximations, obtained by finite difference schemes, of viscosity solutions to the Cauchy problem for first-order Hamilton-Jacobi equations with Neumann type boundary conditions. It states that if Δt is the time-discretization , then the error estimate between the approximation and the solution is of the order [Formula: see text] under certain assumptions of monotonicity and consistency on the numerical scheme.

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