scholarly journals Spectral convergence and nonlinear dynamics of reaction–diffusion equations under perturbations of the domain

2004 ◽  
Vol 199 (1) ◽  
pp. 143-178 ◽  
Author(s):  
Jose M. Arrieta ◽  
Alexandre N. Carvalho
Keyword(s):  
Nonlinear Dynamics ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Spectral Convergence

scholarly journals Nonlinear Dynamics of a Toxin-Phytoplankton-Zooplankton System with Self- and Cross-Diffusion

2016 ◽  
Vol 2016 ◽  
pp. 1-11
Author(s):  
Pengfei Wang ◽  
Min Zhao ◽  
Hengguo Yu ◽  
Chuanjun Dai ◽  
Nan Wang ◽  
...  
Keyword(s):  
Nonlinear Dynamics ◽  
Spatial Distribution ◽  
Theoretical Analysis ◽  
Numerical Simulations ◽  
Reaction Diffusion ◽  
Turing Instability ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Cross Diffusion ◽  
The Rich

A nonlinear system describing the interaction between toxin-producing phytoplankton and zooplankton was investigated analytically and numerically, where the system was represented by a couple of reaction-diffusion equations. We analyzed the effect of self- and cross-diffusion on the system. Some conditions for the local and global stability of the equilibrium were obtained based on the theoretical analysis. Furthermore, we found that the equilibrium lost its stability via Turing instability and patterns formation then occurred. In particular, the analysis indicated that cross-diffusion can play an important role in pattern formation. Subsequently, we performed a series of numerical simulations to further study the dynamics of the system, which demonstrated the rich dynamics induced by diffusion in the system. In addition, the numerical simulations indicated that the direction of cross-diffusion can influence the spatial distribution of the population and the population density. The numerical results agreed with the theoretical analysis. We hope that these results will prove useful in the study of toxic plankton systems.

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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