In the paper we prove  a robust estimate of the Asymptotic Gain type, which characterizes the deviation of solutions of a nonlinear parabolic problem with perturbations on the boundary of the spatial domain from the global attractor of the unperturbed system in terms of the magnitude of the perturbations.

[1] Khalil, Hassan K. Nonlinear systems. Нью-Йорк: Macmillan Publishing Company, 1992.

[2] Sontag E.D. Mathematical control theory: deterministic finite-dimensional systems // Springer. 1998.
[3] Dashkovskiy S., Mironchenko A. Input-to-state stability of infinite dimensional control systems //
Mathematics of Control, Signals and Systems. 2013. Vol. 25, no 1. P. 1–35.
[4] Robinson J.C. Infinite-Dimensional Dynamical Systems // Cambridge University Press. 2001.
[5] R. Temam. Infinite-dimensional dynamical systems in mechanics and physics. Applied Mathematics
Series, Springer, 1997.
[6] A.V. Kapustyan, P.O. Kasyanov, J. Valero. Structure of the global attractor for weak solutions of a
reaction-diffusion equation. Appl. Math. Inform. Sciences, 2015, 9, 2257-2264.
[7] O.V. Kapustyan, N.V Gorban, P.O. Kasyanov. Uniform trajectory attractor for non-autonomous
reaction-diffusion equations with Caratheodory’s nonlinearity. Nonlinear Analysis, 2014, 98, 13-26
[8] P.O. Kasyanov, L. Toscano, N.V. Zadoianchuk. Long-time behaviour of solutions for autonomous
evolution hemivariational inequality with multidimensional reaction-displacement law. Abstract and
Applied Analysis, 2012, 450984.
[9] O.V. Kapustyan, M.O. Perestyuk Global attractors of momentum infinite-dimensional systems, Ukrainian Math. Journal, 68, №4, 517-528 (2016); English translation: Ukr. Math. J., 68, №4, 517-528 (2016)

[10] S. Dashkovskiy, O.A. Kapustian, O.V. Kapustyan, N.V. Gorban. Attractors for Multivalued Impulsive
Systems: Existence and Applications to Reaction-Diffusion System. Mathematical Problems in Engineering,
2021, 7385450
[11] S. Dashkovskiy, O. Kapustyan, J. Schmid A local input-to-state stability result w.r.t. attractors of
nonlinear reaction-diffusion equations. Mathematics of Control, Signals and Systems 80г-є б , 2020, 32,
309–326.
[12] O. Kapustyan, S. Dashkovskiy, J. Schmid. Asymptotic gain results for global attractors of semilinear
systems. Mathematical Control and Related Fields, 2022, 12(3), 763–788
[13] O. Kapustyan, S. Dashkovskiy. Robustness of global attractors: abstract framework and application to
dissipative wave equation. Evolution Equations and Control Theory, 2022, 11(5), 1565-1577
[14] Kapustyan O.V., Krasnieieva A.O. Stability of the global attractor of the reaction-diffusion equation with respect to perturbations at the boundary of the domain // Nonlinear Oscillations, 2024, Vol. 27, No. 2, pp. 229-237.
[15] Vishik M.I., Chepyzhov V.V. Averaging trajectory attractors of evolutionary equations with rapidly
oscillating terms // Sb. Math. 2001. Vol. 192, no 11. P. 13–50.
[16] L.C. Evans. Partial differential eqations. AMS, 1997.