×

zbMATH — the first resource for mathematics

On a global solution and asymptotic behaviour for the generalized damped extensible beam equation. (English) Zbl 0884.35105
The damped extensible beam referred to in the title motivates the study of a fourth order nonlinear PDE on a bounded domain \(\Omega\) with smooth boundary. The author considers the generalized Cauchy problem summarized here:
Let \(A\) be a linear operator in \(H=L^2 (\Omega)\) with dense domain \(V\). Let \(g\) be a non-decreasing continuous scalar function with \(g(0)=0\). Then, for all \(T>0\) and for appropriately regular initial conditions, there is a unique solution \(u\in W^{1,\infty} (0,T;V)\cap W^{2,\infty} (0,T;H)\) to the equation \[ u''+A^2 u+M \bigl(|A^{1/2} u|^2\bigr) Au+g(u')=0 \text{ in } L^{(q+1)/q} (0,T;V'). \] The existence of such a solution is established using the Faedo-Galerkin method, and much of that process is standard. The asymptotic behavior is derived via an energy estimate obtained from repeated (and precise) applications of Young’s Inequality.
The report is well written and easy to follow. The final chapter includes an example using the Laplace operator in \(\mathbb{R}^n\).

MSC:
35L75 Higher-order nonlinear hyperbolic equations
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
35B40 Asymptotic behavior of solutions to PDEs
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Ball, Stability theory for an extensible beam, J. differential equations, 14, 399-418, (1973) · Zbl 0247.73054
[2] Bernstein, I.N., On a class of functional differential equations, Izv. akad. nauk. SSR ser. math., 4, 17-26, (1940)
[3] Biler, Remark on the decay for damped string and beam equations, Nonlinear anal. theory methods appl., 10, 839-842, (1986) · Zbl 0611.35057
[4] Brito, E.H., The damped elastic stretched string equation generalized: existence, uniqueness, regularity and stability, Appl. anal., 13, 219-233, (1982) · Zbl 0458.35065
[5] Brito, E.H., Decay estimates for the generalized damped extensible string and beam equations, Nonlinear anal. theory methods appl., 8, 1489-1496, (1984) · Zbl 0524.35026
[6] Carrier, G.F., On the nonlinear vibration problem of elastic string, Quart. appl. math., 3, 157, (1945) · Zbl 0063.00715
[7] Dafermos; Hrusa, W., Energy methods for quasilinear hyperbolic initial – boundary value problems, Arch. rat. mech. anal., 87, 267-292, (1985) · Zbl 0586.35065
[8] Dickey, R.W., Infinite systems of nonlinear oscillation equations with linear damping, SIAM J. appl. math., 19, 208-214, (1970) · Zbl 0233.34014
[9] Komornik, V., Exact controllability and stabilization – the multiplier method, (1994), John Wiley New York/Masson, Paris · Zbl 0937.93003
[10] Komornik, V., Decay estimates for the wave equation with internal damping, International numerical series analysis, Vol. 118, (1994) · Zbl 0810.35064
[11] Kouémou-Patcheu, S., Existence globale et décroissance exponentielle de l’énergie d’une équation quasi-linéaire, C. R. acad. sci. Paris ser. I, 322, 631-634, (1996) · Zbl 0846.35084
[12] J.-L. Lions, 1968, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod Gauthiers-Villars
[13] Medeiros, L.A., On a new class of nonlinear wave equations, J. math. anal. appl., 69, 252-262, (1979) · Zbl 0407.35051
[14] Nishihara, K., On a global solution of some quasilinear hyperbolic equations, Tokyo J. math., 7, 437-459, (1984) · Zbl 0586.35059
[15] Pohozaev, S.I., On a class of quasilinear hyperbolic equations, Mat. USSR sb., 25, 145-158, (1975) · Zbl 0328.35060
[16] Rivera Rodriguez, P.H., On local strong solutions of a non-linear partial differential equation, Appl. anal., 8, 93-104, (1980) · Zbl 0451.35042
[17] Tucsnak, M., Thèse de doctorat de l’université d’Orléans, (1992)
[18] Woinowsky-Krieger, S., The effect of axial force on the vibration of hinged bars, J. appl. mech., 17, 35-36, (1950) · Zbl 0036.13302
[19] Yamada, Y., On some quasilinear wave equations with dissipative terms, Nagoya math. J., 87, 17-39, (1982) · Zbl 0501.35058
[20] Zuazua, E., Stability and decay estimates for a class of nonlinear hyperbolic problems, Asymptotic anal., 1, 161-185, (1988) · Zbl 0677.35069
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.