Article
Full entry |
PDF
(0.2 MB)
Feedback
PDF
(0.2 MB)
Feedback
Keywords:
functional evolution equation; second order quasilinear equation; monotone operator
functional evolution equation; second order quasilinear equation; monotone operator
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.
References:
[1] Adams, R. A.: Sobolev Spaces. Pure and Applied Mathematics 65. A Series of Monographs and Textbooks Academic Press, New York (1975). MR 0450957 | Zbl 0314.46030
[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. MR 1205967 | Zbl 0806.47055
[3] Czernous, W.: Global solutions of semilinear first order partial functional differential equations with mixed conditions. Funct. Differ. Equ. 18 (2011), 135-154. MR 2894319 | Zbl 1232.35184
[4] Gajewski, H., Gröger, K., Zacharias, K.: Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen. Mathematische Lehrbücher und Monographien II. Abteilung 38 Akademie, Berlin German (1974). MR 0636412
[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. DOI 10.1006/jmaa.2001.7437 | MR 1843969
[6] Hale, J.: Theory of Functional Differential Equations. Applied Mathematical Sciences 3 Springer, New York (1977). DOI 10.1007/978-1-4612-9892-2_3 | MR 0508721 | Zbl 0352.34001
[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. MR 1372735 | Zbl 0840.34083
[8] Lions, J. L.: Quelques méthodes de résolution des problèmes aux limites non linéaires. Études mathématiques Dunod; Gauthier-Villars, Paris French (1969). MR 0259693 | Zbl 0189.40603
[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. MR 2663173 | Zbl 1263.35214
[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. MR 2663173 | Zbl 1263.35214
[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. MR 2508168 | Zbl 1291.35006
[12] Simon, L.: On nonlinear hyperbolic functional differential equations. Math. Nachr. 217 (2000), 175-186. DOI 10.1002/1522-2616(200009)217:1<175::AID-MANA175>3.0.CO;2-N | MR 1780777
[13] Simon, L., Jäger, W.: On non-uniformly parabolic functional differential equations. Stud. Sci. Math. Hung. 45 (2008), 285-300. MR 2417974 | Zbl 1174.35054
[14] Wu, J.: Theory and Applications of Partial Functional-Differential Equations. Applied Mathematical Sciences 119 Springer, New York (1996). DOI 10.1007/978-1-4612-4050-1 | Zbl 0870.35116
[15] Zeidler, E.: Nonlinear Functional Analysis and Its Applications. II/A: Linear Monotone Operators. Springer New York (1990). MR 1033497 | Zbl 0684.47028
[16] Zeidler, E.: Nonlinear Functional Analysis and Its Applications. II/B: Nonlinear Monotone Operators. Springer New York (1990). MR 1033498 | Zbl 0684.47029
Search
Partner of
