Global existence and non-existence problem for a reaction—diffusion equation with a conserved first integral

1995 ◽  
Vol 451 (1943) ◽  
pp. 701-709
Keyword(s):  
Global Existence ◽  
Diffusion Equation ◽  
Finite Time ◽  
Existence Of Solutions ◽  
Reaction Diffusion ◽  
First Integral ◽  
Stationary Solutions ◽  
Reaction Diffusion Equation ◽  
Existence Problem ◽  
Appl Math

In this paper, we prove the global existence and non-existence of solutions of the following problem: RDC{ u t = u xx + u 2 - ∫ u 2 ( x ) d x , x ϵ (0, 1), t > 0, u x (0, t ) = u x (1, t ) = t > 0, u ( x , 0) = u 0 ( x ), x ϵ (0, 1), ∫ 1 0 u ( x, t ) d x = 0, t > 0, Moreover, let u m ( x ) be a stationary solution of problem RDC with m zeros in the interval (0, 1) for m ϵ N , and if we take u 0 ( x ). Then we have that the solution exists globally if 0 < ϵ < 1, and blows up in finite time if ϵ > 1. This result verifies the numerical results of Budd et al . (1993, SIAM Jl appl. Math . 53, 718-742) that the non-zero stationary solutions are unstable.

Remarks on nonlinear elastic waves with null conditions

2020 ◽  
Vol 26 ◽  
pp. 121
Author(s):  
Dongbing Zha ◽  
Weimin Peng
Keyword(s):  
Initial Data ◽  
Global Existence ◽  
Elastic Waves ◽  
Wave Equations ◽  
Alternative Proof ◽  
Classical Solutions ◽  
Small Initial Data ◽  
Hyperelastic Materials ◽  
Variational Structure ◽  
Nonlinear Elastic

For the Cauchy problem of nonlinear elastic wave equations for 3D isotropic, homogeneous and hyperelastic materials with null conditions, global existence of classical solutions with small initial data was proved in R. Agemi (Invent. Math. 142 (2000) 225–250) and T. C. Sideris (Ann. Math. 151 (2000) 849–874) independently. In this paper, we will give some remarks and an alternative proof for it. First, we give the explicit variational structure of nonlinear elastic waves. Thus we can identify whether materials satisfy the null condition by checking the stored energy function directly. Furthermore, by some careful analyses on the nonlinear structure, we show that the Helmholtz projection, which is usually considered to be ill-suited for nonlinear analysis, can be in fact used to show the global existence result. We also improve the amount of Sobolev regularity of initial data, which seems optimal in the framework of classical solutions.

The Affective Politics of Karma among Buddhist Cancer Charities in Vietnam

2020 ◽  
Vol 15 (4) ◽  
pp. 33-62
Author(s):  
Sara Swenson
Keyword(s):  
Public Health ◽  
Global Existence ◽  
Food Production ◽  
Public Health Intervention ◽  
Health Intervention ◽  
Moral Failure ◽  
Cancer Rates ◽  
Charity Work

In this article, I explore how Buddhist charity workers in Vietnam interpret rising cancer rates through understandings of karma. Rather than framing cancer as a primarily physical or medical phenomenon, volunteers state that cancer is a product of collective moral failure. Corruption in public food production is both caused by and perpetuates bad karma, which negatively impacts global existence. Conversely, charity work creates merit, which can improve collective karma and benefit all living beings. I argue that through such interpretations of karma, Buddhist volunteers understand their charity at cancer hospitals as an affective and ethical form of public health intervention.

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