On a stochastic nonlinear equation in one-dimensional viscoelasticity

Kim, Jong

doi:10.1090/S0002-9947-01-02894-X

On a stochastic nonlinear equation in one-dimensional viscoelasticity
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Trans. Amer. Math. Soc. 354 (2002), 1117-1135 Request permission

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.

References

Graham Andrews, On the existence of solutions to the equation $u_{tt}=u_{xxt}+\sigma (u_{x})_{x}$, J. Differential Equations 35 (1980), no. 2, 200–231. MR 561978, DOI 10.1016/0022-0396(80)90040-6
A. Bensoussan and R. Temam, Équations aux dérivées partielles stochastiques non linéaires. I, Israel J. Math. 11 (1972), 95–129 (French). MR 448561, DOI 10.1007/BF02761449
Jöran Bergh and Jörgen Löfström, Interpolation spaces. An introduction, Grundlehren der Mathematischen Wissenschaften, No. 223, Springer-Verlag, Berlin-New York, 1976. MR 0482275
Constantine M. Dafermos, The mixed initial-boundary value problem for the equations of nonlinear one-dimensional viscoelasticity, J. Differential Equations 6 (1969), 71–86. MR 241831, DOI 10.1016/0022-0396(69)90118-1
Giuseppe Da Prato and Jerzy Zabczyk, Stochastic equations in infinite dimensions, Encyclopedia of Mathematics and its Applications, vol. 44, Cambridge University Press, Cambridge, 1992. MR 1207136, DOI 10.1017/CBO9780511666223
A. de Bouard and A. Debussche, On the stochastic Korteweg-de Vries equation, J. Funct. Anal. 154 (1998), no. 1, 215–251 (English, with English and French summaries). MR 1616536, DOI 10.1006/jfan.1997.3184
Daisuke Fujiwara, Concrete characterization of the domains of fractional powers of some elliptic differential operators of the second order, Proc. Japan Acad. 43 (1967), 82–86. MR 216336
James M. Greenberg, Richard C. MacCamy, and Victor J. Mizel, On the existence, uniqueness, and stability of solutions of the equation $\sigma ^{\prime } \,(u_{x})u_{xx}+\lambda u_{xtx}=\rho _{0}u_{tt}$, J. Math. Mech. 17 (1967/1968), 707–728. MR 0225026
J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, Paris; Gauthier-Villars, Paris, 1969 (French). MR 0259693
Michel Métivier and Giovanni Pistone, Sur une équation d’évolution stochastique, Bull. Soc. Math. France 104 (1976), no. 1, 65–85 (French). MR 420854
Michel Métivier and Michel Viot, On weak solutions of stochastic partial differential equations, Stochastic analysis (Paris, 1987) Lecture Notes in Math., vol. 1322, Springer, Berlin, 1988, pp. 139–150. MR 962869, DOI 10.1007/BFb0077872
Étienne Pardoux, Sur des équations aux dérivées partielles stochastiques monotones, C. R. Acad. Sci. Paris Sér. A-B 275 (1972), A101–A103 (French). MR 312572
Jacques Printems, The stochastic Korteweg-de Vries equation in $L^2(\mathbf R)$, J. Differential Equations 153 (1999), no. 2, 338–373. MR 1683626, DOI 10.1006/jdeq.1998.3548
Michel Viot, Solution en loi d’une équation aux dérivées partielles stochastique non linéaire: méthode de compacité, C. R. Acad. Sci. Paris Sér. A 278 (1974), 1185–1188 (French). MR 355349
Michel Viot, Solution en loi d’une équation aux dérivées partielles stochastique non linéaire: méthode de compacité, C. R. Acad. Sci. Paris Sér. A 278 (1974), 1185–1188 (French). MR 355349
Viot, M., Solutions faibles aux équations aux dérivées partielles stochastiques non linéaires, Thèse, Université Pierre et Marie Curie, Paris (1976).
John B. Walsh, An introduction to stochastic partial differential equations, École d’été de probabilités de Saint-Flour, XIV—1984, Lecture Notes in Math., vol. 1180, Springer, Berlin, 1986, pp. 265–439. MR 876085, DOI 10.1007/BFb0074920
Richard Kershner, The number of circles covering a set, Amer. J. Math. 61 (1939), 665–671. MR 43, DOI 10.2307/2371320