Boutet de Monvel, L.; Chueshov, I. D.; Rezounenko, A. V. Long-time behaviour of strong solutions of retarded nonlinear P. D. E. s. (English) Zbl 0891.35159 Commun. Partial Differ. Equations 22, No. 9-10, 1453-1474 (1997). 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) Cited in 21 Documents MSC: 35R10 Partial functional-differential equations 35L75 Higher-order nonlinear hyperbolic equations 35B40 Asymptotic behavior of solutions to PDEs Keywords:existence; long-time behaviour; finite-dimensional attractor; continuous dependence on the parameters PDFBibTeX XMLCite \textit{L. Boutet de Monvel} et al., Commun. Partial Differ. Equations 22, No. 9--10, 1453--1474 (1997; Zbl 0891.35159) Full Text: DOI 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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.