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Long-time behaviour of strong solutions of retarded nonlinear P. D. E. s. (English) Zbl 0891.35159
The paper deals with the following retarded PDE: $\mu\ddot u+ \gamma\dot u+ \Delta^2u- f\Biggl(\int_\Omega|\nabla u(x,t)|^2dx\Biggr) \Delta u+ \rho{\partial u\over\partial x_1}- q(u_t)= p_0(x),\quad x\in\Omega,\quad t>0,$ with boundary conditions $u\bigl|_{t=0+}= u_0,\quad \dot u\bigl|_{t=0+}= u_1,\quad u\bigl|_{t\in(- t_*,0)}= \varphi(x, t)$ and initial conditions $u\bigl|_{\partial\Omega}= \Delta u\bigl|_{\partial\Omega}= 0,$ where $$\Omega\subset\mathbb{R}^n$$ is a bounded domain, $$u_t= u_t(\theta)= u(t+\theta)$$, $$\theta\in (-t_*,0)$$ is the retarded function. The authors prove existence of strong solutions and investigate the long-time behaviour of these solutions. The existence of a finite-dimensional attractor is proved as well as its continuous dependence on the parameters of the system.
Reviewer: D.Bainov (Sofia)

MSC:
 35R10 Functional partial differential equations 35L75 Higher-order nonlinear hyperbolic equations 35B40 Asymptotic behavior of solutions to PDEs
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References:
 [1] Babin, A.V. and Vishik, M.I. 1992. ”Attractors of Evolutionary Equations”. North-Holland:Amsterdam. · Zbl 0778.58002 [2] Berger M., J. Appl. Mech 22 pp 465– (1955) [3] Boutet de Monvel L., Annali di Mat. pura ed applicata 171 (1997) [4] Boutet L., C.R. Acad. Sci. Paris. SerI 322 pp 1001– (1996) [5] DOI: 10.1007/BF01097291 · Zbl 0783.73046 · doi:10.1007/BF01097291 [6] DOI: 10.1070/RM1993v048n03ABEH001033 · doi:10.1070/RM1993v048n03ABEH001033 [7] Chueshov I.D., Lecture Notes (1991) [8] Chueshov I.D., Math Notes 47 pp 401– (1990) [9] Chueshov I.D., Math. Physics, Analysis, Geometry 2 pp 363– (1995) [10] Dmitrieva Zh.N., J. of Soviet Mathematics 22 pp 48– (1978) [11] Dowell, E.H. 1975. ”Aeroelastisity of Plates and Shells”. Leyden:Noordhoff International Publishing. · Zbl 0306.73039 [12] Hale J.K., Amer. Math. Soc (1998) [13] Hale, J.K. ”Theory of Functional Differential Equations”. · Zbl 1092.34500 [14] DOI: 10.1016/0005-1098(78)90036-5 · Zbl 0385.93028 · doi:10.1016/0005-1098(78)90036-5 [15] Kapitanskii L.V., Leningrad Math J 2 pp 97– (1991) [16] Krasil’shchikova, E.A. 1978. ”The Thin Wing in a Compressible Flow”. Nauka, In Russian [17] DOI: 10.1070/RM1987v042n06ABEH001503 · Zbl 0687.35072 · doi:10.1070/RM1987v042n06ABEH001503 [18] Lions, J.L. 1969. ”Quelques Méthods de Résolutions des Problémes aux Limites Non Linéaires”. Paris:Dunod. [19] Rezounenko A.V., Math. Physics, Analysis, Geometry 4 (1997) [20] Sevcovic D., Commenr, Math. Univ. Carolinae 31 pp 283– (1990) [21] DOI: 10.1007/BF01762360 · Zbl 0629.46031 · doi:10.1007/BF01762360 [22] Temam, R. 1988. ”Infinite Dimensional Dynamic Systems in Mechanics and Physics”. Berlin, New York:Springer. · Zbl 0662.35001
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