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Remark on the decay for damped string and beam equations. (English) Zbl 0611.35057

It is shown that the norms of the solutions to nonlinear weakly damped hyperbolic equations like \[ u''(t)+\alpha A^ 2u(t)+\delta u'(t)+M(| A^{1/2}u(t)|^ 2)Au(t)=0 \] decay exponentially to zero when t tends to infinity.
The obtained decay rates are optimal - the same as for the linear equation with \(M\equiv const.\)
Some more precise estimates for strongly damped equations with the damping term of type Au’ will be proved in a forthcoming paper in Nonlinear Analysis (1987).

MSC:

35L20 Initial-boundary value problems for second-order hyperbolic equations
35B40 Asymptotic behavior of solutions to PDEs
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
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References:

[1] De Brito, E. H., Decay estimates for the generalized damped extensible string and beam equations, Nonlinear Analysis, 8, 1489-1496 (1984) · Zbl 0524.35026
[2] Moore, R. A.; Nehari, Z., Nonoscillation theorems for a class of nonlinear diffential equations, Trans. Am. math. Soc., 93, 30-52 (1959) · Zbl 0089.06902
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