Asymptotic Behavior of Solution for Coupled Reaction Diffusion System by Order m

Maroua, Mebarki; Nabila, Barrouk

doi:10.28924/2291-8639-21-2023-37

Research Article

Asymptotic Behavior of Solution for Coupled Reaction Diffusion System by Order m

Mebarki Maroua
Barrouk Nabila¹*

a:1:{s:5:"en_US";s:159:"Faculty of Science and Technology, Department of Mathematics and Computer Science, Mohamed Cherif Messaadia University, P.O.Box 1553, Souk Ahras 41000, Algeria";}

* Corresponding author

Abstract

The aim of this paper is to prove that asymptotic behavior in the time of solutions for the weakly coupled reaction diffusion system:
∂u_i/∂t − d_i∆u_i = f_i (u₁, u₂, …, u_m) in Ω×R⁺,
∂u_i/∂η = 0 in ∂Ω×R⁺, (0.1)
u_i(., 0) = u_i⁰(.) in Ω,
where Ω is an open bounded domain of class C¹ in Rⁿ, u_i(t, x), i=1, m, t≥0, x∈Ω are real valued functions. We treat the system (0.1) as a dynamical system in C(Ω) × C(Ω) × ... × C(Ω) and apply Lyapunov type stability techniques. A key ingredient in this analysis is a result which establishes that the orbits of the dynamical system are precompact in C(Ω) × C(Ω) × ... × C(Ω). As a consequence of Arzela-Ascoli theorem, this will be satisfied if the orbits are, for example, uniformly bounded in C¹(Ω) × C¹(Ω) × ... × C¹(Ω) for t>0.

How to Cite

M. Maroua, B. Nabila, Asymptotic Behavior of Solution for Coupled Reaction Diffusion System by Order m, Int. J. Anal. Appl. 21 (2023), 37. https://doi.org/10.28924/2291-8639-21-2023-37

BibTeX

@article{Maroua2023,
  author  = {Maroua, Mebarki and Nabila, Barrouk},
  title   = {Asymptotic Behavior of Solution for Coupled Reaction Diffusion System by Order m},
  journal = {Int. J. Anal. Appl.},
  volume  = {21},
  year    = {2023},
  pages   = {37},
  doi     = {10.28924/2291-8639-21-2023-37}
}

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