×

zbMATH — the first resource for mathematics

Existence and decay of solutions of an abstract second order nonlinear problem. (English) Zbl 1179.34064
This paper is concerned with the global existence and behavior of solutions to initial value problems of the form
\[ u''(t)+Au(t)+B^\alpha u'(t) =0; \quad u(0)=u_0, u'(0)=u_1, \] where \(A\) is a monotone nonlinear operator, \(B\) is a linear operator and \(\alpha\) is a real number with \(0<\alpha \leq 1\). The conditions are stated in terms of the monotonicity, differentiability and other properties of \(A\).

MSC:
34G20 Nonlinear differential equations in abstract spaces
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Berkovits, J., A note on the embedding theorem of Browder and ton, Proc. amer. math. soc., 131, 2963-2966, (2003) · Zbl 1023.47029
[2] Brezis, H., Analyse fonctionnelle, théorie et applications, (1993), Masson Paris
[3] Browder, F.E.; Ton, Bui An, Nonlinear functional equations in Banach spaces and elliptic super-regularization, Math. Z., 105, 177-195, (1968) · Zbl 0165.49802
[4] Greenberg, J.M.; MacCamy, R.C.; Mizel, V.J., On the existence, uniqueness, and stability of solutions of the equation \(\sigma^\prime(u_x) u_x + \lambda u_{x t x} = \rho_0 u_{t t}\), J. math. mech., 17, 707-728, (1968) · Zbl 0157.41003
[5] Greenberg, J.M., On the existence, uniqueness, and stability of solutions of the equation \(\rho_0 X_{t t} = E(X_x) X_{x x} + \lambda X_{x x t}\), J. math. anal. appl., 25, 575-591, (1969) · Zbl 0192.44803
[6] Lions, J.L., Quelques Méthodes de Résolution des problèmes aux limites non linéaires, (1969), Dunod Paris · Zbl 0189.40603
[7] Milla Miranda, M., Análise espectral EM espaços de Hilbert, Textos de Métodos matemáticos, vol. 28, (1994), IM-UFRJ Rio
[8] Nakao, M., Decay of solutions of some nonlinear evolution equations, J. math. anal. appl., 60, 542-549, (1977) · Zbl 0376.34051
[9] Tsutsumi, M., Some nonlinear evolution equations of second order, Proc. Japan acad., 47, 950-955, (1971) · Zbl 0258.35017
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.