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Zbl 0704.45013
Pereira, Ducival Carvalho
Existence, uniqueness and asymptotic behavior for solutions of the nonlinear beam equation.
(English)
[J] Nonlinear Anal., Theory Methods Appl. 14, No.8, 613-623 (1990). ISSN 0362-546X

Of concern is the existence, uniqueness and asymptotic behavior of solutions to an initial value problem for the second-order equation $$Ku''(t)+A\sp 2u(t)+M(\vert A\sp{1/2}u(t)\vert\sp 2)\quad Au(t)+u'(t)=0,\quad t\ge 0,$$ in a Hilbert space H. Here $\vert \cdot \vert$ denotes the norm in H, K: $H\to H$ is linear monotone, A is a given linear unbounded operator of H, and M is a real function on $[0,\infty)$. The author extends earlier results by {\it P. Biler} [ibid. 10, 839-842 (1986; Zbl 0611.35057)], and {\it E. H. Brito} [ibid. 8, 1489-1496 (1984; Zbl 0524.35026); ibid. 11, 125-137 (1987; Zbl 0613.34013)].
[S.Aizicovici]
MSC 2000:
*45N05 Integral equations in abstract spaces
45K05 Integro-partial differential equations
34G20 Nonlinear ODE in abstract spaces
35L70 Second order nonlinear hyperbolic equations

Keywords: nonlinear beam equation; existence; uniqueness; asymptotic behavior; initial value problem; second-order equation; Hilbert space; linear unbounded operator

Citations: Zbl 0611.35057; Zbl 0524.35026; Zbl 0613.34013

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