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On a stochastic nonlinear equation in one-dimensional viscoelasticity


Author: Jong Uhn Kim
Journal: Trans. Amer. Math. Soc. 354 (2002), 1117-1135
MSC (2000): Primary 35R60, 60H15, 74D10
DOI: https://doi.org/10.1090/S0002-9947-01-02894-X
Published electronically: October 4, 2001
MathSciNet review: 1867374
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Abstract: In this paper we discuss an initial-boundary value problem for a stochastic nonlinear equation arising in one-dimensional viscoelasticity. We propose to use a new direct method to obtain a solution. This method is expected to be applicable to a broad class of nonlinear stochastic partial differential equations.


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  • [1] Andrews, G., On the existence of solutions to the equation $u_{tt} = u_{xxt} + \sigma (u_{x})_{x}$, J. Diff. 35 (1980), pp. 200 - 231. MR 81d:35073
  • [2] Bensoussan, A. and Temam, R., Équations aux dérivées partielles stochastiques non linéaires, Israel J. Math. 11 (1972), pp. 95 - 129. MR 56:6867
  • [3] Bergh, J. and Löfström, J., Interpolation Spaces, Springer-Verlag, Berlin-Heidelberg-New York, 1976. MR 58:2349
  • [4] Dafermos, C.M., The mixed initial-boundary value problem for the equations of nonlinear one-dimensional viscoelasticity, J. Diff. Equations 6 (1969), pp. 71 - 86. MR 39:3168
  • [5] Da Prato, G. and Zabczyk, J., Stochastic Equations in Infinite Dimensions, Cambridge University Press, 1992. MR 95g:60073
  • [6] de Bouard, A. and Debussche, A., On the stochastic Korteweg-de Vries equation, J. Func. Analysis 154 (1998), pp. 215 - 251. MR 99c:35209
  • [7] Fujiwara, D., Concrete characterization of the domains of fractional powers of some elliptic differential operators of the second order, Proc. Japan Acad. 43 (1967), pp. 82 - 86. MR 35:7170
  • [8] Greenberg, J.M., MacCamy, R.C. and Mizel, V.J., On the existence, uniqueness and stability of solutions of the equation $\sigma '(u_{x}) u_{xx} + \lambda u_{xtx} = \rho _{0} u_{tt}$, J. Math. Mech. 17 (1968), pp. 707 - 728. MR 37:623
  • [9] Lions, J.L., Quelques méthodes de résolution des problémes aux limites non linéaires, Dunod, Paris, 1969. MR 41:4326
  • [10] Métivier, M. and Pistone, G., Sur une équation d'évolution stochastique, Bull. Soc. Math. France 104 (1976), pp. 65 - 85. MR 54:8866
  • [11] Métivier, M. and Viot, M., On weak solutions of stochastic partial differential equations, Lecture Notes in Math. No.1322, Springer-Verlag, 1987, pp. 139 - 150. MR 90a:60113
  • [12] Pardoux, E., Sur des équations aux dérivées partielles stochastiques monotones, C.R.Acad. Sci. Paris, série A 275 (1972), pp. 101 - 103. MR 47:1129
  • [13] Printems, J., The stochastic Korteweg-de Vries equation in $L^{2}(R)$, J. Diff. Equations 153 (1999), pp. 338 - 373. MR 2001g:35280
  • [14] Viot, M., Solution en loi d'une équation aux dérivées partiellles stochastique non linéaire: méthode de compacité, C.R. Acad. Sci. Paris, série A 278 (1974), pp. 1185 - 1188. MR 50:7824a
  • [15] Viot, M., Solution en loi d'une équation aux dérivées partiellles stochastique non linéaire: méthode de monotonie, C.R. Acad. Sci. Paris, série A 278 (1974), pp. 1405 - 1408. MR 50:7824b
  • [16] Viot, M., Solutions faibles aux équations aux dérivées partielles stochastiques non linéaires, Thèse, Université Pierre et Marie Curie, Paris (1976).
  • [17] Walsh, J.B., An introduction to stochastic partial differential equations, Lecture Notes in Math., No.1180, Springer-Verlag, 1984, pp. 266 - 437. MR 88a:60114
  • [18] Yamada, T. and Watanabe, S., On the uniqueness of solutions of stochastic differential equations, J. Math. Kyoto Univ. 11 (1971), pp. 155 - 167, pp. 553 - 563. MR 43:4150; MR 44:6071

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Additional Information

Jong Uhn Kim
Affiliation: Department of Mathematics, Virginia Tech, Blacksburg, Virginia 24061-0123
Email: kim@math.vt.edu

DOI: https://doi.org/10.1090/S0002-9947-01-02894-X
Keywords: Viscoelasticity, random force, white noise, pathwise solutions
Received by editor(s): October 19, 2000
Received by editor(s) in revised form: May 4, 2001
Published electronically: October 4, 2001
Article copyright: © Copyright 2001 American Mathematical Society

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