Existence, multiplicity, shape and attractivity of heterogeneous steady states for bistable reaction-diffusion equations in the plane

2019 ◽  
Vol 267 (7) ◽  
pp. 4014-4046
Author(s):  
Taishan Yi ◽  
Xingfu Zou
Keyword(s):  
Reaction Diffusion ◽  
Steady States ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations

A mathematical framework for determining the stability of steady states of reaction–diffusion equations with periodic source terms

2019 ◽  
Vol 84 (4) ◽  
pp. 669-678
Author(s):  
Lennon Ó Náraigh ◽  
Khang Ee Pang
Keyword(s):  
Steady State ◽  
A Priori ◽  
Reaction Diffusion ◽  
Spatial Dimension ◽  
Steady States ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Mathematical Framework ◽  
Source Terms ◽  
The Stability

Abstract We develop a mathematical framework for determining the stability of steady states of generic nonlinear reaction–diffusion equations with periodic source terms in one spatial dimension. We formulate an a priori condition for the stability of such steady states, which relies only on the properties of the steady state itself. The mathematical framework is based on Bloch’s theorem and Poincaré’s inequality for mean-zero periodic functions. Our framework can be used for stability analysis to determine the regions in an appropriate parameter space for which steady-state solutions are stable.

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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