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Rate of convergence for reaction–diffusion equations with nonlinear Neumann boundary conditions and \({\mathcal {C}}^1\) variation of the domain

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Abstract

In this paper, we propose the compact convergence approach to deal with the continuity of attractors of some reaction–diffusion equations under smooth perturbations of the domain subject to nonlinear Neumann boundary conditions. We define a family of invertible linear operators to compare the dynamics of perturbed and unperturbed problems in the same phase space. All continuity arising from small smooth perturbations will be estimated by a rate of convergence given by the domain variation in a \({\mathcal {C}}^1\) topology.

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Acknowledgements

The first author (MCP) is partially supported by CNPq 308950/2020-8, FAPESP 2020/04813-0 and 2020/14075-6 (Brazil). The second author (LP) acknowledges the hospitality of the Applied Mathematics Department at IME-USP where part of this work was done. We would also like to thank the anonymous referee for the comments and suggestions, which improved our paper.

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Authors and Affiliations

  1. University of São Paulo, São Paulo, SP, Brazil

    Marcone C. Pereira

  2. State University of Ponta Grossa, Ponta Grossa, PR, Brazil

    Leonardo Pires

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  1. Marcone C. Pereira
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  2. Leonardo Pires
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Correspondence to Marcone C. Pereira.

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Pereira, M.C., Pires, L. Rate of convergence for reaction–diffusion equations with nonlinear Neumann boundary conditions and \({\mathcal {C}}^1\) variation of the domain. J. Evol. Equ. 24, 5 (2024). https://doi.org/10.1007/s00028-023-00934-7

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