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Dynamics of Fractional Delayed Reaction-Diffusion Equations.

Linfang Liu1, Juan J Nieto2

  • 1Department of Mathematics, Northwest University, Xi'an 710127, China.

Entropy (Basel, Switzerland)
|June 28, 2023
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Summary
This summary is machine-generated.

This study investigates the long-term behavior of weak solutions for fractional delayed reaction-diffusion equations. We prove the existence and uniqueness of solutions and determine the global attracting set.

Keywords:
bounded variable delayfractional reaction–diffusion equationsgeneralized comparison principalgeneralized fractional derivativeglobal attracting sets

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Area of Science:

  • Partial Differential Equations
  • Fractional Calculus
  • Dynamical Systems

Background:

  • Reaction-diffusion equations model phenomena across various scientific fields.
  • Fractional derivatives offer more complex modeling capabilities than traditional integer-order derivatives.
  • Delayed differential equations are crucial for systems with time lags.

Purpose of the Study:

  • To analyze the long-term behavior of weak solutions for fractional delayed reaction-diffusion equations.
  • To establish the existence and uniqueness of these weak solutions.
  • To determine the global attracting set for the dynamical system.

Main Methods:

  • Utilized the Galerkin approximation method for solution analysis.
  • Employed the comparison principle to prove existence and uniqueness.
  • Applied Sobolev embedding theorems and Halanay inequality to find the global attracting set.

Main Results:

  • Proved the existence and uniqueness of weak solutions for the specified equation.
  • Successfully obtained the global attracting set of the system.
  • Demonstrated the effectiveness of combining Galerkin approximation with analytical inequalities.

Conclusions:

  • The study provides a rigorous mathematical framework for understanding fractional delayed reaction-diffusion systems.
  • The findings contribute to the theoretical understanding of complex dynamical systems with memory effects.
  • This work lays the foundation for further research into the qualitative behavior of such models.