Stability Theory for Multiple Equilibrium States of a Nonlinear Diffusion Process: A Singularly Perturbed Eigenvalue Problem

1973 ◽  
Vol 4 (1) ◽  
pp. 134-140 ◽  
Author(s):  
Herbert B. Keller
Keyword(s):  
Eigenvalue Problem ◽  
Diffusion Process ◽  
Stability Theory ◽  
Nonlinear Diffusion ◽  
Equilibrium States ◽  
Singularly Perturbed ◽  
Multiple Equilibrium

scholarly journals A Weak Limit Theorem for Generalized Jiřina Processes

2009 ◽  
Vol 46 (2) ◽  
pp. 453-462 ◽  
Author(s):  
Yuqiang Li
Keyword(s):  
Limit Theorem ◽  
Diffusion Process ◽  
Nonlinear Diffusion ◽  
Weak Limit ◽  
Weak Limit Theorem ◽  
Lévy Jumps

In this paper we prove that a sequence of scaled generalized Jiřina processes can converge weakly to a nonlinear diffusion process with Lévy jumps under certain conditions.

scholarly journals Global and Local Structures of Bifurcation Curves of ODE with Nonlinear Diffusion

2018 ◽  
Vol 2018 ◽  
pp. 1-7
Author(s):  
Tetsutaro Shibata
Keyword(s):  
Mathematical Model ◽  
Eigenvalue Problem ◽  
Nonlinear Diffusion ◽  
Nonlinear Term ◽  
Nonlinear Eigenvalue Problem ◽  
Maximum Norm ◽  
Bifurcation Parameter ◽  
Local Structures ◽  
Global And Local ◽  
The Mathematical Model

We consider the nonlinear eigenvalue problem Duu′′+λfu=0, u(t)>0, t∈I≔(0,1), u(0)=u(1)=0, where D(u)=uk, f(u)=u2n-k-1+sin⁡u, and λ>0 is a bifurcation parameter. Here, n∈N and k (0≤k<2n-1) are constants. This equation is related to the mathematical model of animal dispersal and invasion, and λ is parameterized by the maximum norm α=uλ∞ of the solution uλ associated with λ and is written as λ=λ(α). Since f(u) contains both power nonlinear term u2n-k-1 and oscillatory term sin⁡u, it seems interesting to investigate how the shape of λ(α) is affected by f(u). The purpose of this paper is to characterize the total shape of λ(α) by n and k. Precisely, we establish three types of shape of λ(α), which seem to be new.

scholarly journals Selective Sharpening of Color Images through Coupled Nonlinear Diffusion Process

2004 ◽  
Vol 58 (11) ◽  
pp. 1673-1679
Author(s):  
Takahiro Saito ◽  
Hiroyuki Harada ◽  
Jun Satsumabayashi ◽  
Takashi Komatsu
Keyword(s):  
Diffusion Process ◽  
Nonlinear Diffusion ◽  
Color Images
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