Long time behavior of solutions for a scalar nonlocal reaction-diffusion equation

2011 ◽  
Vol 96 (5) ◽  
pp. 483-490 ◽  
Author(s):  
Xiaoliu Wang ◽  
Weifeng Wo
Keyword(s):  
Diffusion Equation ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Long Time ◽  
Nonlocal Reaction

scholarly journals On a model of a population with variable motility

2015 ◽  
Vol 25 (10) ◽  
pp. 1961-2014 ◽  
Author(s):  
Olga Turanova
Keyword(s):  
Diffusion Equation ◽  
Long Range ◽  
Viscosity Solutions ◽  
Jacobi Equation ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Reaction Term ◽  
Long Time ◽  
Nonlocal Reaction ◽  
Hamilton Jacobi Equation

We study a reaction–diffusion equation with a nonlocal reaction term that models a population with variable motility. We establish a global supremum bound for solutions of the equation. We investigate the asymptotic (long-time and long-range) behavior of the population. We perform a certain rescaling and prove that solutions of the rescaled problem converge locally uniformly to zero in a certain region and stay positive (in some sense) in another region. These regions are determined by two viscosity solutions of a related Hamilton–Jacobi equation.

scholarly journals Blow-Up and Global Existence of Solutions for the Time Fractional Reaction–Diffusion Equation

Mathematics ◽  
2021 ◽  
Vol 9 (24) ◽  
pp. 3248
Author(s):  
Linfei Shi ◽  
Wenguang Cheng ◽  
Jinjin Mao ◽  
Tianzhou Xu
Keyword(s):  
Initial Data ◽  
Diffusion Equation ◽  
Blow Up ◽  
Sufficient Conditions ◽  
Global Solutions ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Time Behavior ◽  
Global Existence Of Solutions ◽  
Long Time

In this paper, we investigate a reaction–diffusion equation with a Caputo fractional derivative in time and with boundary conditions. According to the principle of contraction mapping, we first prove the existence and uniqueness of local solutions. Then, under some conditions of the initial data, we obtain two sufficient conditions for the blow-up of the solutions in finite time. Moreover, the existence of global solutions is studied when the initial data is small enough. Finally, the long-time behavior of bounded solutions is analyzed.

Long-time behavior of solutions and chaos in reaction-diffusion equations

2017 ◽  
Vol 99 ◽  
pp. 91-100 ◽  
Author(s):  
Kamal N. Soltanov ◽  
Anatolij K. Prykarpatski ◽  
Denis Blackmore
Keyword(s):  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Long Time
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