# Article

 Title: Second order quasilinear functional evolution equations (English) Author: Simon, László Language: English Journal: Mathematica Bohemica ISSN: 0862-7959 (print) ISSN: 2464-7136 (online) Volume: 140 Issue: 2 Year: 2015 Pages: 139-152 Summary lang: English . Category: math . Summary: We consider second order quasilinear evolution equations where also the main part contains functional dependence on the unknown function. First, existence of solutions in $(0,T)$ is proved and examples satisfying the assumptions of the existence theorem are formulated. Then a uniqueness theorem is proved. Finally, existence and some qualitative properties of the solutions in $(0,\infty )$ (boundedness and stabilization as $t\to \infty$) are shown. (English) Keyword: functional evolution equation Keyword: second order quasilinear equation Keyword: monotone operator MSC: 35A01 MSC: 35A02 MSC: 35B35 MSC: 35R10 MSC: 35R20 idZBL: Zbl 06486930 idMR: MR3368490 DOI: 10.21136/MB.2015.144322 . Date available: 2015-06-30T12:13:51Z Last updated: 2020-07-29 Stable URL: http://hdl.handle.net/10338.dmlcz/144322 . Reference: [1] Adams, R. A.: Sobolev Spaces.Pure and Applied Mathematics 65. A Series of Monographs and Textbooks Academic Press, New York (1975). Zbl 0314.46030, MR 0450957 Reference: [2] Berkovits, J., Mustonen, V.: Topological degree for perturbations of linear maximal monotone mappings and applications to a class of parabolic problems.Rend. Mat. Appl., VII. Ser. 12 (1992), 597-621. Zbl 0806.47055, MR 1205967 Reference: [3] Czernous, W.: Global solutions of semilinear first order partial functional differential equations with mixed conditions.Funct. Differ. Equ. 18 (2011), 135-154. Zbl 1232.35184, MR 2894319 Reference: [4] Gajewski, H., Gröger, K., Zacharias, K.: Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen.Mathematische Lehrbücher und Monographien II. Abteilung 38 Akademie, Berlin German (1974). MR 0636412 Reference: [5] Giorgi, C., Rivera, J. E. Muñoz, Pata, V.: Global attractors for a semilinear hyperbolic equation in viscoelasticity.J. Math. Anal. Appl. 260 (2001), 83-99. MR 1843969, 10.1006/jmaa.2001.7437 Reference: [6] Hale, J.: Theory of Functional Differential Equations.Applied Mathematical Sciences 3 Springer, New York (1977). Zbl 0352.34001, MR 0508721, 10.1007/978-1-4612-9892-2_3 Reference: [7] Hartung, F., Turi, J.: Stability in a class of functional-differential equations with state-dependent delays.Qualitative Problems for Differential Equations and Control Theory World Scientific Singapore (1995), 15-31 C. Corduneanu. Zbl 0840.34083, MR 1372735 Reference: [8] Lions, J. L.: Quelques méthodes de résolution des problèmes aux limites non linéaires.Études mathématiques Dunod; Gauthier-Villars, Paris French (1969). Zbl 0189.40603, MR 0259693 Reference: [9] Simon, L.: Nonlinear functional parabolic equations.Integral Methods in Science and Engineering: Computational Methods 2. Selected papers based on the presentations at the 10th International Conference on Integral Methods in Science and Engineering, Santander, Spain, 2008 Birkhäuser Boston (2010), 321-326 C. Constanda et al. Zbl 1263.35214, MR 2663173 Reference: [10] Simon, L.: On nonlinear functional parabolic equations with state-dependent delays of Volterra type.Int. J. Qual. Theory Differ. Equ. Appl. 4 (2010), 88-103. Zbl 1263.35214, MR 2663173 Reference: [11] Simon, L.: Application of monotone type operators to parabolic and functional parabolic {PDE}'s.Handbook of Differential Equations: Evolutionary Equations IV Elsevier/North-Holland, Amsterdam (2008), 267-321 C. M. Dafermos et al. Zbl 1291.35006, MR 2508168 Reference: [12] Simon, L.: On nonlinear hyperbolic functional differential equations.Math. Nachr. 217 (2000), 175-186. MR 1780777, 10.1002/1522-2616(200009)217:1<175::AID-MANA175>3.0.CO;2-N Reference: [13] Simon, L., Jäger, W.: On non-uniformly parabolic functional differential equations.Stud. Sci. Math. Hung. 45 (2008), 285-300. Zbl 1174.35054, MR 2417974 Reference: [14] Wu, J.: Theory and Applications of Partial Functional-Differential Equations.Applied Mathematical Sciences 119 Springer, New York (1996). Zbl 0870.35116, 10.1007/978-1-4612-4050-1 Reference: [15] Zeidler, E.: Nonlinear Functional Analysis and Its Applications. II/A: Linear Monotone Operators.Springer New York (1990). Zbl 0684.47028, MR 1033497 Reference: [16] Zeidler, E.: Nonlinear Functional Analysis and Its Applications. II/B: Nonlinear Monotone Operators.Springer New York (1990). Zbl 0684.47029, MR 1033498 .

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