Abstract
We analyze the long-time behavior of the semigroup S(t) generated by a heat equation with degenerate past history and a nonlinear heat supply posed in a three dimensional bounded domain \(\Omega \). Assuming that the degeneracy occurs in a positive measure subset of \(\Omega \), we prove the existence and regularity of the global attractor associated to this semigroup.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alabau-Boussouira, F., Cannarsa, P., Fragnelli, G.: Carleman estimates for degenerate parabolic operators with applications to null controllability. J. Evol. Equ. 6, 161–204 (2006)
Barbu, V.: Nonlinear semigroups and differential equations in Banach spaces. Noordhoff International Publishing, Groningen (1976)
Boutaayamou, I., Fragnelli, G., Maniar, L.: Carleman estimates for parabolic equations with interior degeneracy and Neumann boundary conditions. J. d’Anal. Math. 135, 1–35 (2018)
Campiti, M., Metafune, G., Pallara, D.: Degenerate self-adjoint evolution equations on the unit interval. Semigroup Forum 57, 1–36 (1998)
Cannarsa, P., Martinez, P., Vancostenoble, J.: Carleman estimates for a class of degenerate parabolic operators. SIAM J. Control Optim. 47, 1–19 (2008)
Cannarsa, P., Tort, J., Yamamoto, M.: Unique continuation and approximate controllability for a degenerate parabolic equation. Appl. Anal. 91, 1409–1425 (2012)
Cavalcanti, M.M., Domingos Cavalcanti, V.N., Jorge Silva, M.A., de Souza Franco, A.Y.: Exponential stability for the wave model with localized memory in a past history framework. J. Differ. Equ. 264, 6535–6584 (2018)
Chepyzhov, V.V., Mainini, E., Pata, V.: Stability of abstract linear semigroups arising from heat conduction with memory. Asymptot. Anal. 50(3–4), 269–291 (2006)
Chueshov, I., Eller, M., Lasiecka, I.: On the attractor for a semilinear wave equation with critical exponent and nonlinear boundary dissipation. Commun. PDE 27, 1901–1951 (2002)
Conti, M., Pata, V., Squassina, M.: Singular limit of differential systems with memory. Indiana Univ. Math. J. 55, 169–215 (2006)
Conti, M., Pata, V.: On the regularity of global attractors. Discrete Contin. Dyn. Syst. 25(4), 1209–1217 (2009)
Conti, M., Marchini, E.M., Pata, V.: Nonclassical diffusion with memory. Math. Methods Appl. Sci. 38, 948–958 (2015)
Dafermos, C.M.: Asymptotic stability in viscoelasticity. Arch. Ration. Mech. Anal. 37, 297–308 (1970)
Faria, J.C.O.: Carleman estimates and observability inequalities for a class of problems ruled by parabolic equations with interior degenaracy. Appl. Math. Optim. (2020). https://doi.org/10.1007/s00245-019-09651-5
Fragnelli, G., Goldstein, G.R., Goldstein, J.A., Romanelli, S.: Generators with interior degeneracy on spaces of \(L^2\) type. Electron J. Differ. Equ. 189, 1–30 (2012)
Fragnelli, G., Mugnai, D.: Carleman estimates and observability inequalities for parabolic equation with interior degeneracy. Adv. Nonlinear Anal. 2, 339–378 (2013)
Fragnelli, G., Mugnai, D.: Carleman estimates, observability inequalities and null controlability for interior degenerate non smooth parabolic equations. American Mathematical Society, Providence (2016)
Gao, H., Li, L., Liu, Z.: Stability of degenerate heat equation in non-cylindrical/cylindrical domain. Z. Angew. Math. Phys. 70(14), 1–17 (2019). https://doi.org/10.1007/s00033-019-1166-3
Giorgi, C., Marzocchi, A., Pata, V.: Asymptotic behavior of a semilinear problem in heat conduction with memory. Nonlinear Differ. Equ. Appl. 5, 333–354 (1998)
Giorgi, C., Marzocchi, A., Pata, V.: Uniform attractors for a non-autonomous semilinear heat equation with memory. Quart. Appl. Math. 58, 661–683 (2000)
Giorgi, C., Naso, M.G., Pata, V.: Exponential stability in linear heat conduction with memory: a semigroup approach. Commun. Appl. Anal. 5, 121–134 (2001)
Giorgi, C., Pata, V.: Asymptotic behavior of a nonlinear hyperbolic heat equation with memory. NoDEA Nonlinear Differ. Equ. Appl. 8, 157–171 (2001)
Grasselli, M., Pata, V.: A reaction-diffusion equation with memory. Discrete Contin. Dyn. Syst. 15, 1079–1088 (2006)
Gurtin, M.E., Pipkin, A.C.: A general theory of heat conduction with finite wave speeds. Arch. Ration. Mech. Anal. 31, 113–126 (1968)
Lions, J.L., Magenes, E.: Non-Homogeneous Boundary Value Problems and Applications vol I. In: Die Grundlehren der Mathematischen Wissenschaften. Springer-Verlag, New York (1972)
Showalter, R.E.: Monotone Operators in Banach Space and Nonlinear Partial Differential Equations. American Mathematical Society, Providence (1997)
Temam, R.: Infinit Dimensional Dynamical System in Mechanichs and Physics. Springer, New York (1997)
Wang, C.: Boundary behavior and asymptotic behavior of solutions to a class of parabolic equations with boundary degeneracy. Discrete Contin. Dyn. Syst. 36, 1041–1060 (2016)
Wang, X., Duan, F., Hu, D.: Attractors for a class of abstract evolution equations with fading memory. Math. Probl. Eng. Article ID 1410408, 16 pages (2017)
Acknowledgements
The authors would like to thank the anonymous referee for helpful suggestions that significantly contributed to improving the initial version of the paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Faria, J.C.O., Webler, C.M. Existence of a Global Attractor for the Heat Equation with Degenerate Memory. J Dyn Diff Equat 35, 845–864 (2023). https://doi.org/10.1007/s10884-021-10042-0
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10884-021-10042-0