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On the decay of solutions for some nonlinear evolution equations of second order

Published online by Cambridge University Press:  22 January 2016

Yoshio Yamada
Yoshio Yamada*
Affiliation:
Department of Mathematics, Faculty of Science, Nagoya University
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In this paper we consider nonlinear evolution equations of the form

(E) u″(t) + Au(t) + B(t)u′(t) = f(t), 0 ≦ t < ∞,

(u′(t) = d2u(t)/dt2, u′(t) = du(t)/dt), where A and B(t) are (possibly) nonlinear operators. Various examples of equations of type (E) arise in physics; for instance, if Au = –Δu and B(t)u′ = | u′ | u′, the equation represents a classical vibrating membrane with the resistance proportional to the velocity.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 73 , March 1979 , pp. 69 - 98
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1979

References

[1] Ball, J. M., Initial-boundary value problems for an extensible beam, J. Math. Anal. Appl. 42 (1973), 61–90.CrossRefGoogle Scholar
[2] Ball, J. M., Stability theory for an extensible beam, J. Diff. Eqns. 14 (1973), 399–418.CrossRefGoogle Scholar
[3] Ball, J. M. and Carr, J., Decay to zero in critical cases of second order ordinary differential equations of Duffing type, Arch. Rat. Mech. Anal. 63 (1976), 47–57.CrossRefGoogle Scholar
[4] Barbu, V., Nonlinear semigroups and differential equations in Banach spaces, Noordhoff (1976).CrossRefGoogle Scholar
[5] Brezis, H., Problèmes unilatéraux, J. Math, pures et appl. 51 (1972), 1–168.Google Scholar
[6] Caughey, T. K. and Ellison, J., Existence, uniqueness and stability of solutions of a class of nonlinear partial differential equations, J. Math. Anal. Appl. 51 (1975), 1–32.CrossRefGoogle Scholar
[7] Greenberg, J. M., On the existence, uniqueness, and stability of solutions of the equation J. Math. Anal. Appl. 25 (1969), 575–591.CrossRefGoogle Scholar
[8] Greenberg, J. M., MacCamy, R. C. and Mizel, V. J., On the existence, uniqueness, and stability of solutions of the equation J. Math. Mech. 17 (1968), 707–728.Google Scholar
[9] Levine, H. A. and Murray, A. C., Asymptotic behavior and lower bounds for semi-linear wave equations in Hilbert space with applications, SIAM J. Math. Anal. 6 (1975), 846–859.CrossRefGoogle Scholar
[10] Lions, J. L., Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod Gauthier-Villars (1969).Google Scholar
[11] Lions, J. L. and Strauss, W. A., Some non-linear evolution equations, Bull. Soc. math. France 93 (1965), 43–96.CrossRefGoogle Scholar
[12] Nakao, M., Convergence of solutions of the wave equation with a nonlinear dissipative term to the steady state, Memoirs Fac. Sci. Kyushu Univ. 30 (1976), 257–265.Google Scholar
[13] Nakao, M., Asymptotic stability of the bounded or almost periodic solution of the wave equation with nonlinear dissipative term, J. Math. Anal. Appl. 58 (1977), 336–343.CrossRefGoogle Scholar
[14] Nakao, M., Decay of solutions of some nonlinear evolution equations, J. Math. Anal. Appl. 60 (1977), 542–549.CrossRefGoogle Scholar
[15] Nakao, M., A difference inequality and its application to nonlinear evolution equations, J. Math. Soc. Japan 30 (1978), 747–762.CrossRefGoogle Scholar
[16] Sattinger, D. H., Stability of nonlinear hyperbolic equations, Arch. Rat. Mech. Anal. 28 (1968), 226–244.CrossRefGoogle Scholar
[17] Strauss, W. A., On continuity of functions with values in various Banach spaces, Pacific J. Math. 19 (1966), 543–551.CrossRefGoogle Scholar
[18] Tsutsumi, M., Some nonlinear evolution equations of second order, Proc. Japan Acad. 47 (1971), 950–955.CrossRefGoogle Scholar
[19] Yamada, Y., Note on certain nonlinear evolution equations of second order, to appear.Google Scholar