Geodesic
Global and blowup solutions of a mixed problem with nonlinear boundary conditions for a one-dimensional semilinear wave equation
Sbornik. Mathematics, Tome 205 (2014) no. 4, pp. 573-599 Cet article a éte moissonné depuis la source Math-Net.Ru

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A mixed problem for a one-dimensional semilinear wave equation with nonlinear boundary conditions is considered. Conditions of this type occur, for example, in the description of the longitudinal oscillations of a spring fastened elastically at one end, but not in accordance with Hooke's linear law. Uniqueness and existence questions are investigated for global and blowup solutions to this problem, in particular how they depend on the nature of the nonlinearities involved in the equation and the boundary conditions. Bibliography: 14 titles.
Keywords: semilinear wave equation, nonlinear boundary conditions, a priori estimate, comparison theorems, global and blowup solutions. 
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S. S. Kharibegashvili; O. M. Jokhadze. Global and blowup solutions of a mixed problem with nonlinear boundary conditions for a one-dimensional semilinear wave equation. Sbornik. Mathematics, Tome 205 (2014) no. 4, pp. 573-599. http://geodesic.mathdoc.fr/item/SM_2014_205_4_a5/

[1] A. N. Tikhonov, A. A. Samarskii, Equations of mathematical physics, The Macmillan Co., New York, 1963, xvi+765 pp. | MR | MR | Zbl | Zbl

[2] V. B. Kolmanovskii, V. R. Nosov, Ustoichivost i periodicheskie rezhimy reguliruemykh sistem s posledeistviem, Teoreticheskie osnovy kibernetiki, Nauka, M., 1981, 448 pp. | MR | Zbl

[3] S. A. Rodrìguez, J.-M. Dion, L. Dugard, “Stability of neutral time delay systems: a survey of some results”, Advances in automatic control, Kluwer Internat. Ser. Engrg. Comput. Sci., 754, Kluwer Acad. Publ., Boston, MA, 2004, 315–335 | DOI | MR | Zbl

[4] A. S. Ackleh, Keng Deng, “Existence and nonexistence of global solutions of the wave equation with a nonlinear boundary condition”, Quart. Appl. Math., 59:1 (2001), 153–158 | MR | Zbl

[5] E. Vitillaro, “Global existence for the wave equation with nonlinear boundary damping and source term”, J. Differential Equations, 186:1 (2002), 259–298 | DOI | MR

[6] J. Serrin, G. Todorova, E. Vitillaro, “Existence for a nonlinear wave equation with damping and source terms”, Differential Integral Equations, 16:1 (2003), 13–50 | MR | Zbl

[7] Hongwei Zhang, Qingying Hu, “Asymptotic behaviour and nonexistence of wave equation with nonlinear boundary condition”, Commun. Pure Appl. Anal., 4:4 (2005), 861–896 | DOI | MR | Zbl

[8] A. Nowakowski, “Solvability and stability of a semilinear wave equation with nonlinear boundary conditions”, Nonlinear Anal., 73:6 (2010), 1495–1514 | DOI | MR | Zbl

[9] V. P. Mikhaĭlov, Partial differential equations, Mir, M.; distributed by Imported Publications, Inc., 1978, 397 pp. | MR | MR | Zbl | Zbl

[10] V. S. Vladimirov, Equations of mathematical physics, Pure and Applied Mathematics, 3, Marcel Dekker, Inc., New York, 1971, vi+418 pp. | MR | MR | Zbl | Zbl

[11] R. Narasimhan, Analysis on real and complex manifolds, Adv. Stud. Pure Math., 1, Masson Cie, Éditeurs, Paris; North-Holland Publishing Co., Amsterdam, 1968, x+246 pp. | MR | Zbl | Zbl

[12] D. Gilbarg, N. S. Trudinger, Elliptic partial differential equations of second order, Grundlehren Math. Wiss., 224, 2nd ed., Springer-Verlag, Berlin, 1983, xiii+513 pp. | MR | MR | Zbl | Zbl

[13] V. A. Trenogin, Funktsionalnyi analiz, 2-e izd., Nauka, M., 1993, 440 pp. | MR | Zbl

[14] E. Mitidieri, S. I. Pokhozhaev, “A priori estimates and blow-up of solutions to nonlinear partial differential equations and inequalities”, Proc. Steklov Inst. Math., 234 (2001), 1–362 | MR | Zbl