Bulletin of the Korean Mathematical Society (대한수학회보)

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WELL-POSEDNESS AND ASYMPTOTIC BEHAVIOR OF PARTLY DISSIPATIVE REACTION DIFFUSION SYSTEMS WITH MEMORY

  • Vu Trong Luong (VNU-University of Education Vietnam National University) ;
  • Nguyen Duong Toan (Faculty of Mathematics and Natural Sciences Haiphong University)
  • Received : 2023.02.09
  • Accepted : 2023.03.17
  • Published : 2024.01.31
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Abstract

In this paper, we consider the asymptotic behavior of solutions for the partly dissipative reaction diffusion systems of the FitzHugh-Nagumo type with hereditary memory and a very large class of nonlinearities, which have no restriction on the upper growth of the nonlinearity. We first prove the existence and uniqueness of weak solutions to the initial boundary value problem for the above-mentioned model. Next, we investigate the existence of a uniform attractor of this problem, where the time-dependent forcing term h ∈ L2b(ℝ; H-1(ℝN)) is the only translation bounded instead of translation compact. Finally, we prove the regularity of the uniform attractor A, i.e., A is a bounded subset of H2(ℝN) × H1(ℝN) × L2µ(ℝ+, H2(ℝN)). The results in this paper will extend and improve some previously obtained results, which have not been studied before in the case of non-autonomous, exponential growth nonlinearity and contain memory kernels.

Keywords

Acknowledgement

The authors would like to thank the reviewers for the helpful comments and suggestions which improved the presentation of the paper. This work was completed while the author was visiting the Vietnam Institute of Advanced Study in Mathematics (VIASM). The authors would like to thank the Institute for its hospitality.

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