scholarly journals Blow-up and global existence for the non-local reaction diffusion problem with time dependent coefficient

2013 ◽  
Vol 2013 (1) ◽  
Author(s):  
Iftikhar Ahmed ◽  
Chunlai Mu ◽  
Pan Zheng ◽  
Fuchen Zhang
Keyword(s):  
Global Existence ◽  
Blow Up ◽  
Local Reaction ◽  
Reaction Diffusion ◽  
Diffusion Problem ◽  
Time Dependent ◽  
Non Local

scholarly journals Time-dependent attractors for non-autonomous non-local reaction–diffusion equations

2018 ◽  
Vol 148 (5) ◽  
pp. 957-981
Author(s):  
Tomás Caraballo ◽  
Marta Herrera-Cobos ◽  
Pedro Marín-Rubio
Keyword(s):  
Existence And Uniqueness ◽  
Local Reaction ◽  
Reaction Diffusion ◽  
Strong Solutions ◽  
Reaction Diffusion Equations ◽  
Time Dependent ◽  
Reaction Diffusion Equation ◽  
Pullback Attractors ◽  
Non Local ◽  
Bounded Sets

In this paper the existence and uniqueness of weak and strong solutions for a non-autonomous non-local reaction–diffusion equation is proved. Furthermore, the existence of minimal pullback attractors in the L2-norm in the frameworks of universes of fixed bounded sets and those given by a tempered growth condition is established, along with some relationships between them. Finally, we prove the existence of minimal pullback attractors in the H1-norm and study relationships among these new families and those given previously in the L2 context. We also present new results in the autonomous framework that ensure the existence of global compact attractors as a particular case.

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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