Asymptotic Behavior of Solutions to the Diffusion Equation

2018 ◽  
Vol 49 (4) ◽  
pp. 601-620
Author(s):  
Satyanarayana Engu ◽  
Ahmed Mohd ◽  
Manas Ranjan Sahoo
Keyword(s):  
Asymptotic Behavior ◽  
Diffusion Equation ◽  
Asymptotic Behavior Of Solutions

Well-Posedness and Asymptotic Behavior of a Nonclassical Nonautonomous Diffusion Equation with Delay

2015 ◽  
Vol 25 (14) ◽  
pp. 1540021 ◽  
Author(s):  
Tomás Caraballo ◽  
Antonio M. Márquez-Durán ◽  
Felipe Rivero
Keyword(s):  
Fixed Point ◽  
Asymptotic Behavior ◽  
Diffusion Equation ◽  
Existence Of Solutions ◽  
Local Solution ◽  
Asymptotic Behavior Of Solutions ◽  
Well Posedness ◽  
Pullback Attractors ◽  
Nonautonomous Dynamical Systems ◽  
Equation With Delay

In this paper, a nonclassical nonautonomous diffusion equation with delay is analyzed. First, the well-posedness and the existence of a local solution is proved by using a fixed point theorem. Then, the existence of solutions defined globally in future is ensured. The asymptotic behavior of solutions is analyzed within the framework of pullback attractors as it has revealed a powerful theory to describe the dynamics of nonautonomous dynamical systems. One difficulty in the case of delays concerns the phase space that one needs to construct the evolution process. This yields the necessity of using a version of the Ascoli–Arzelà theorem to prove the compactness.

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