scholarly journals Invariant regions for systems of lattice reaction–diffusion equations

2017 ◽  
Vol 263 (11) ◽  
pp. 7601-7626 ◽  
Author(s):  
Antonín Slavík
Keyword(s):  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Invariant Regions

scholarly journals Nonnegative solutions to the reaction-diffusion equations for prey-predator models with the dormancy of predators

2021 ◽  
Author(s):  
Novrianti Novrianti ◽  
Okihiro Sawada ◽  
Naoki Tsuge
Keyword(s):  
A Priori ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Classical Solutions ◽  
Evolution Operators ◽  
A Priori Bounds ◽  
Nonnegative Solutions ◽  
Invariant Regions ◽  
Global Unique Solvability

The time-global unique solvability on the reaction–diffusion equations for preypredator models and dormancy on predators is established. The crucial step is to construct time-local nonnegative classical solutions by using a new approximation associated with time-evolution operators. Although the system does not equip usual comparison principles, a priori bounds are derived, so solutions are extended time-globally. Via observations to the corresponding ordinary differential equations, invariant regions and asymptotic behaviors of solutions are also investigated.

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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