Nonclassical symmetry solutions for reaction–diffusion equations with explicit spatial dependence

2007 ◽  
Vol 67 (9) ◽  
pp. 2541-2552 ◽  
Author(s):  
B.H. Bradshaw-Hajek ◽  
M.P. Edwards ◽  
P. Broadbridge ◽  
G.H. Williams
Keyword(s):  
Spatial Dependence ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Nonclassical Symmetry

scholarly journals Nonclassical Symmetry Solutions for Non-Autonomous Reaction-Diffusion Equations

Symmetry ◽  
2019 ◽  
Vol 11 (2) ◽  
pp. 208 ◽  
Author(s):  
Bronwyn Bradshaw-Hajek
Keyword(s):  
Spatial Dependence ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Invariant Solutions ◽  
Group Invariant ◽  
Nonlinear Reaction ◽  
Nonclassical Symmetry ◽  
Group Invariant Solutions ◽  
Dynamics Modelling

The behaviour of many systems in chemistry, combustion and biology can be described using nonlinear reaction diffusion equations. Here, we use nonclassical symmetry techniques to analyse a class of nonlinear reaction diffusion equations, where both the diffusion coefficient and the coefficient of the reaction term are spatially dependent. We construct new exact group invariant solutions for several forms of the spatial dependence, and the relevance of some of the solutions to population dynamics modelling is discussed.

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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