Blowup properties for several diffusion systems with localised sources

2006 ◽  
Vol 48 (1) ◽  
pp. 37-56 ◽  
Author(s):  
Zhaoyin Xiang ◽  
Qiong Chen ◽  
Chunlai Mu
Keyword(s):  
Cauchy Problem ◽  
Boundary Conditions ◽  
Neumann Boundary ◽  
Parabolic Systems ◽  
Neumann Boundary Conditions ◽  
Nonlocal Problems ◽  
Blowup Rate ◽  
Diffusion Systems ◽  
Blowup Criterion ◽  
The Cauchy Problem

AbstractThis paper investigates the Cauchy problem for two classes of parabolic systems with localised sources. We first give the blowup criterion, and then deal with the possibilities of simultaneous blowup or non-simultaneous blowup under some suitable assumptions. Moreover, when simultaneous blowup occurs, we also establish precise blowup rate estimates. Finally, using similar ideas and methods, we shall consider several nonlocal problems with homogeneous Neumann boundary conditions.

scholarly journals Analysis of Dirichlet–Robin Iterations for Solving the Cauchy Problem for Elliptic Equations

2020 ◽  
Author(s):  
Pauline Achieng ◽  
Fredrik Berntsson ◽  
Jennifer Chepkorir ◽  
Vladimir Kozlov
Keyword(s):  
Cauchy Problem ◽  
Boundary Conditions ◽  
Helmholtz Equation ◽  
Iterative Algorithm ◽  
Elliptic Equations ◽  
Numerical Experiments ◽  
Neumann Boundary ◽  
Neumann Boundary Conditions ◽  
General Elliptic ◽  
The Cauchy Problem

Abstract The Cauchy problem for general elliptic equations of second order is considered. In a previous paper (Berntsson et al. in Inverse Probl Sci Eng 26(7):1062–1078, 2018), it was suggested that the alternating iterative algorithm suggested by Kozlov and Maz’ya can be convergent, even for large wavenumbers $$k^2$$ k 2 , in the Helmholtz equation, if the Neumann boundary conditions are replaced by Robin conditions. In this paper, we provide a proof that shows that the Dirichlet–Robin alternating algorithm is indeed convergent for general elliptic operators provided that the parameters in the Robin conditions are chosen appropriately. We also give numerical experiments intended to investigate the precise behaviour of the algorithm for different values of $$k^2$$ k 2 in the Helmholtz equation. In particular, we show how the speed of the convergence depends on the choice of Robin parameters.

scholarly journals Multiple solutions for a class of bi-nonlocal problems with nonlinear Neumann boundary conditions

2022 ◽  
Vol 40 ◽  
pp. 1-11
Author(s):  
Ghasem A. Afrouzi ◽  
Z. Naghizadeh ◽  
Nguyen Thanh Chung
Keyword(s):  
Boundary Conditions ◽  
Variational Methods ◽  
Weak Solutions ◽  
Multiple Solutions ◽  
Neumann Boundary ◽  
Neumann Boundary Conditions ◽  
Nonlocal Problems ◽  
Mapping Theory ◽  
Nonlinear Neumann Boundary Conditions

In this paper, we are interested in a class of bi-nonlocal problems with nonlinear Neumann boundary conditions and sublinear terms at infinity. Using $(S_+)$ mapping theory and variational methods, we establish the existence of at least two non-trivial weak solutions for the problem provied that the parameters are large enough. Our result complements and improves some previous ones for the superlinear case when the Ambrosetti-Rabinowitz type conditions are imposed on the nonlinearities.

Stability Criterion For Linear Oscillatory Parabolic Systems

1992 ◽  
Vol 114 (1) ◽  
pp. 175-178 ◽  
Author(s):  
Keum S. Hong ◽  
Joseph Bentsman
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Numerical Simulations ◽  
Stability Criterion ◽  
Neumann Boundary ◽  
Parabolic Systems ◽  
Neumann Boundary Conditions ◽  
Distributed Parameter ◽  
Parabolic Partial Differential Equations

This paper presents a stability criterion for a class of distributed parameter systems governed by linear oscillatory parabolic partial differential equations with Neumann boundary conditions. The results of numerical simulations that support the criterion are presented as well.

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