Self-Assembly and Beyond

2019 ◽  
pp. 326-350
Author(s):  
Troy Shinbrot
Keyword(s):  
Limit Cycle ◽  
Limit Cycles ◽  
Self Assembly ◽  
Wave Solution ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Turing Patterns ◽  
Limit Cycle Oscillations

Effects of combining reaction with diffusion are examined, and the resulting self-assembly of ordered patterns is overviewed. Turing patterns and limit cycle oscillations are shown to result from these considerations, and future avenues for research into these topics are briefly discussed. Additional topics include reaction-diffusion equations, and limit cycles wave solution, and the limit cycle.

scholarly journals Blow-up analyses in nonlocal reaction diffusion equations with time-dependent coefficients under Neumann boundary conditions

2020 ◽  
Vol 18 (1) ◽  
pp. 1552-1564
Author(s):  
Huimin Tian ◽  
Lingling Zhang
Keyword(s):  
Boundary Conditions ◽  
Blow Up ◽  
Sufficient Conditions ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Neumann Boundary Conditions ◽  
Reaction Diffusion Equations ◽  
Time Dependent ◽  
Diffusion Equations ◽  
Nonlocal Reaction

Abstract In this paper, the blow-up analyses in nonlocal reaction diffusion equations with time-dependent coefficients are investigated under Neumann boundary conditions. By constructing some suitable auxiliary functions and using differential inequality techniques, we show some sufficient conditions to ensure that the solution u ( x , t ) u(x,t) blows up at a finite time under appropriate measure sense. Furthermore, an upper and a lower bound on blow-up time are derived under some appropriate assumptions. At last, two examples are presented to illustrate the application of our main results.

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