# American Institute of Mathematical Sciences

doi: 10.3934/dcdss.2021140
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## Asymptotic behavior of nonlocal partial differential equations with long time memory

 1 Departmento de Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Universidad de Sevilla, c/ Tarfia s/n, 41012 Sevilla, Spain 2 Centro de Investigación Operativa, Universidad Miguel Hernández de Elche, Avda. de la Universidad, s/n, 03202-Elche, Spain

* Corresponding author: Tomás Caraballo

Dedicated to Georg Hetzer on occasion of his 75th birthday

Received  June 2021 Revised  September 2021 Early access November 2021

Fund Project: The research has been partially supported by the Spanish Ministerio de Ciencia, Innovación y Universidades (MCIU), Agencia Estatal de Investigación (AEI) and Fondo Europeo de Desarrollo Regional (FEDER) under the project PGC2018-096540-B-I00, and by Junta de Andalucía (Consejería de Economía y Conocimiento) and FEDER under projects US-1254251 and P18-FR-4509

In this paper, it is first addressed the well-posedness of weak solutions to a nonlocal partial differential equation with long time memory, which is carried out by exploiting the nowadays well-known technique used by Dafermos in the early 70's. Thanks to this Dafermos transformation, the original problem with memory is transformed into a non-delay one for which the standard theory of autonomous dynamical system can be applied. Thus, some results about the existence of global attractors for the transformed problem are {proved}. Particularly, when the initial values have higher regularity, the solutions of both problems (the original and the transformed ones) are equivalent. Nevertheless, the equivalence of global attractors for both problems is still unsolved due to the lack of enough regularity of solutions in the transformed problem. It is therefore proved the existence of global attractors of the transformed problem. Eventually, it is highlighted how to proceed to obtain meaningful results about the original problem, without performing any transformation, but working directly with the original delay problem.

Citation: Jiaohui Xu, Tomás Caraballo, José Valero. Asymptotic behavior of nonlocal partial differential equations with long time memory. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2021140
##### References:
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##### References:
 [1] T. Caraballo, M. J. Garrido-Atienza, B. Schmalfußand and J. Valero, Global attractor for a non-autonomous integro-differential equation in materials with memory, Nonlinear Anal., 73 (2010), 183-201.  doi: 10.1016/j.na.2010.03.012.  Google Scholar [2] T. Caraballo and J. Real, Attractors for 2D-Navier-Stokes models with delays, J. Differential Equations, 205 (2004), 271-297.  doi: 10.1016/j.jde.2004.04.012.  Google Scholar [3] M. Chipot and J. F. Rodrigues, On a class of nonlocal nonlinear elliptic problems, RAIRO Modél. Math. Anal. Numér., 26 (1992), 447-467.  doi: 10.1051/m2an/1992260304471.  Google Scholar [4] M. Chipot and T. Savistka, Nonlocal $p$-Laplace equations depending on the $L^p$ norm of the gradient, Adv. Differential Equations, 19 (2014), 997-1020.   Google Scholar [5] M. Chipot, V. Valente and G. V. Caffarelli, Remarks on a nonlocal problem involving the Dirichlet energy, Rend. Semin. Mat. Univ. Padova, 110 (2003), 199-220.   Google Scholar [6] M. Conti, V. Pata and M. Squassina, Singular limit of differential systems with memory, Indiana Univ. Math. J., 1 (2006), 169-215.  doi: 10.1512/iumj.2006.55.2661.  Google Scholar [7] C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal., 37 (1970), 297-308.  doi: 10.1007/BF00251609.  Google Scholar [8] J. Furter and M. Grinfeld, Local vs. nonlocal interactions in population dynamics, J. Math. Biol., 27 (1989), 65-80.  doi: 10.1007/BF00276081.  Google Scholar [9] S. Gatti, M. Grasselli and V. Pata, Exponential attractors for a phased-field model with memory and quadratic nonlinearities, Indiana Univ. Math. J., 53 (2004), 719-753.  doi: 10.1512/iumj.2004.53.2413.  Google Scholar [10] C. Giorgi, V. Pata and A. Marzochi, Asymptotic behavior of a semilinear problem in heat conduction with memory, NoDEA Nonlinear Differential Equations Appl., 5 (1998), 333-354.  doi: 10.1007/s000300050049.  Google Scholar [11] M. Grasselli and V. Pata, Uniform attractors of nonautonomous dynamical systems with memory, Progr. Nonlinear Differential Equations Appl., 50 (2002), 155-178.   Google Scholar [12] J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Applied Mathematical Sciences, 99. Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7.  Google Scholar [13] Y. J. Li and Y. J. Wang, The existence and asymptotic behavior of solutions to fractional stochastic evolution equations with infinite delay, J. Differential Equations, 266 (2019), 3514-3558.  doi: 10.1016/j.jde.2018.09.009.  Google Scholar [14] J. L. Lions, Quelques Méthodes de Résolutions des Problèm aux Limites Non Linéaires, Dunod Gauthier-Villars, Paris, 1969.  Google Scholar [15] V. Pata and A. Zucchi, Attractors for a damped hyperbolic equation with linear memory, Adv. Math. Sci. Appl., 11 (2001), 505-529.   Google Scholar [16] R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977.  Google Scholar [17] J. H. Xu, Z. C. Zhang and T. Caraballo, Non-autonomous nonlocal partial differential equations with delay and memory, J. Differential Equations, 270 (2021), 505-546.  doi: 10.1016/j.jde.2020.07.037.  Google Scholar
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