Some Results on Reaction Diffusion Equations with Initial Time Difference

2001 ◽  
pp. 391-399
Author(s):  
Aghalaya S. Vatsala
Keyword(s):  
Reaction Diffusion ◽  
Time Difference ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Initial Time

Attractors with higher-order regularity of stochastic reaction–diffusion equations on time-varying domains

2020 ◽  
Vol 20 (02) ◽  
pp. 2050041
Author(s):  
Lu Yang ◽  
Meihua Yang ◽  
Peter Kloeden
Keyword(s):  
Reaction Diffusion ◽  
Higher Order ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Initial Time ◽  
Regular Space ◽  
Time Varying ◽  
Regularity Properties ◽  
Random Attractors ◽  
Stochastic Reaction

Random attractors and their higher-order regularity properties are studied for stochastic reaction–diffusion equations on time-varying domains. Some new a priori estimates for the difference of solutions near the initial time and the continuous dependence in initial data in [Formula: see text] are proved. Then attraction of the random attractors in the higher integrability space [Formula: see text] for any [Formula: see text] and the regular space [Formula: see text] is established.

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.

scholarly journals Characterization of Cocycle Attractors for Nonautonomous Reaction–Diffusion Equations

2016 ◽  
Vol 26 (08) ◽  
pp. 1650135 ◽  
Author(s):  
C. A. Cardoso ◽  
J. A. Langa ◽  
R. Obaya
Keyword(s):  
Chaotic Dynamics ◽  
Linear Part ◽  
Reaction Diffusion ◽  
Positive Cone ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Minimal Set ◽  
Full Characterization ◽  
The Stability

In this paper, we describe in detail the global and cocycle attractors related to nonautonomous scalar differential equations with diffusion. In particular, we investigate reaction–diffusion equations with almost-periodic coefficients. The associated semiflows are strongly monotone which allow us to give a full characterization of the cocycle attractor. We prove that, when the upper Lyapunov exponent associated to the linear part of the equations is positive, the flow is persistent in the positive cone, and we study the stability and the set of continuity points of the section of each minimal set in the global attractor for the skew product semiflow. We illustrate our result with some nontrivial examples showing the richness of the dynamics on this attractor, which in some situations shows internal chaotic dynamics in the Li–Yorke sense. We also include the sublinear and concave cases in order to go further in the characterization of the attractors, including, for instance, a nonautonomous version of the Chafee–Infante equation. In this last case we can show exponentially forward attraction to the cocycle (pullback) attractors in the positive cone of solutions.

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