Existence, uniqueness and uniform decay for the nonlinear beam degenerate equation with weak damping. (English) Zbl 1063.74067

Summary: We prove global existence and uniqueness of weak solutions of the problem for the nonlinear beam degenerate equation \[ K(x, t)u''+ \Delta^2u+ M\Biggl(\int_\Omega |\nabla u|^2 dx\Biggr)(-\Delta u)+ u'= 0\quad\text{in }Q= \Omega\times (0,\infty), \] where \(Q\) is a cylindrical domain of \(\mathbb{R}^{n+1}\), \(n\geq 1\), with the lateral boundary \(\Sigma\), and \(K(x,t)\) is a real function defined in \(Q\), \(K(x,t)\geq 0\) for all \((x,t)\in\Omega\times (0,\infty)\), which satisfies some appropriate conditions. \(M(\lambda)\) is a real function such that \(M(\lambda)\geq -\beta\), \(0< \beta< \lambda_1\), \(\lambda_1\) is the first eigenvalue of the operator \(\Delta^2\). Moreover, the uniform decay rates of the energy are obtained as time goes to infinity.


74K10 Rods (beams, columns, shafts, arches, rings, etc.)
74H20 Existence of solutions of dynamical problems in solid mechanics
74H25 Uniqueness of solutions of dynamical problems in solid mechanics
74H40 Long-time behavior of solutions for dynamical problems in solid mechanics
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[1] Ball, J.M., Initial-boundary value problem for an extensible beam, J. math. anal. appl., 42, 61-90, (1973) · Zbl 0254.73042
[2] Ball, J.M., Stability theory for an extensible beam, J. differ. equ., 14, 399-418, (1973) · Zbl 0247.73054
[3] Biller, P., Remark on the decay for generalized damped string and beam equations, Nonlin. anal., 10, 839-842, (1986) · Zbl 0611.35057
[4] Brito, E.H., Decay estimates for generalized damped extensible string and beam equation, Nonlin. anal., 8, 1489-1496, (1984) · Zbl 0524.35026
[5] Burgreen, D., Free vibrations of a pinended column with constant distance between pinendes, J. appl. mech., 18, 135-139, (1951)
[6] Dickey, R.W., Free vibrations and dynamics buckling of extensible beam, J. math. anal. appl., 29, 443-454, (1970) · Zbl 0187.04803
[7] Eisley, J.G., Nonlinear vibrations of beams and rectangular plates, Z. angew. math. phys., 15, 167-175, (1964) · Zbl 0133.19101
[8] J. Ferreira, Sobre uma equação hiperbólica – parabólica semilinear, Doctoral Thesis, IM-UFRJ, 1991
[9] Ferreira, J., On weak solutions of a nonlinear hyperbolic – parabolic partial differential equation, Comput. appl. math., 14, 3, 269-283, (1995) · Zbl 0854.35073
[10] Ferreira, J., On weak solutions of semilinear hyperbolic – parabolic equations, Int. J. math. math. sci., 19, 751-758, (1996) · Zbl 0861.35062
[11] J.L. Lions, Quelques Méthodos de Résolution des Problèmes aux Limites non Linéaires, Dunod, Paris, 1969 · Zbl 0189.40603
[12] Medeiros, L.A., On a new class of nonlinear wave equations, J. math. appl., 69, 252-262, (1979) · Zbl 0407.35051
[13] Menzala, G.P., On classical solutions of a quasilinear hyperbolic equations, Nonlin. anal., 3, 613-627, (1978) · Zbl 0419.35062
[14] Mikhlin, S.G., Variational methods in mathematical physics, (1964), Pergamon Press Oxford · Zbl 0119.19002
[15] Nakao, M., Decay of solutions for some nonlinear evolution equations, J. math. anal. appl., 60, 542-546, (1977)
[16] Pereira, D.C., Existence, uniqueness and asymptotic behavior for solutions of the nonlinear beam equation, Nonlin. anal., 14, 8, 613-623, (1990) · Zbl 0704.45013
[17] D.C. Pereira, Existência, unicidade e comportamento assintóico das soluções da equação não-linear da viga, Doctoral Thesis, IM-UFRJ, 1987
[18] Ramos, O.C., Regularity property for the nonlinear beam operator, An. acad. bras. cienc., 61, 1, 15-24, (1989) · Zbl 0702.35052
[19] Rivera, J.E.M., Existence and asymptotic behaviour in a class of nonlinear wave equation with thermal dissipation, Nonlin. anal., 23, 10, 1255-1271, (1994)
[20] Woinowsky-Krieger, S., The effect of axial force on the vibration of hinged bars, J. appl. mech., 17, 35-36, (1950) · Zbl 0036.13302
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