Proceedings of the Japan Academy, Series A, Mathematical Sciences

Existence result for a doubly degenerate quasilinear stochastic parabolic equation

Mamadou Sango

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Abstract

Using the splitting-up method, we establish a new existence result for an initial boundary value problem for the doubly degenerate stochastic quasilinear parabolic equation \begin{equation*} d \bigl( \left|y\right|^{\alpha-2}y\bigr) -\left[ \sum_{i=1}^{n} \frac{\partial}{\partial x_{i}} \left( \left| \frac{\partial y}{\partial x}\right|^{p-2} \frac{\partial y}{\partial x_{i}}\right) - g(t,y) \right] dt = \sum_{l=0}^{d}h_{l} (t,y) dW_{t}^{l}, \end{equation*} where $W_{t}^{l}$ are one-dimensional Wiener process defined on a complete probability space, $p$, $\alpha$ and the functions $g$ and $h_{l}$ satisfy appropriate restrictions.

Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 81, Number 5 (2005), 89-94.

Dates
First available in Project Euclid: 3 June 2005

Permanent link to this document
https://projecteuclid.org/euclid.pja/1117805148

Digital Object Identifier
doi:10.3792/pjaa.81.89

Mathematical Reviews number (MathSciNet)
MR2143549

Zentralblatt MATH identifier
1330.35554

Subjects
Primary: 60H15: Stochastic partial differential equations [See also 35R60] 35R60: Partial differential equations with randomness, stochastic partial differential equations [See also 60H15]

Keywords
Doubly degenerate stochastic parabolic equations splitting-up method compactness

Citation

Sango, Mamadou. Existence result for a doubly degenerate quasilinear stochastic parabolic equation. Proc. Japan Acad. Ser. A Math. Sci. 81 (2005), no. 5, 89--94. doi:10.3792/pjaa.81.89. https://projecteuclid.org/euclid.pja/1117805148


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References

  • A. Bamberger, Étude d'une équation doublement non linéaire, J. Functional Analysis 24 (1977), no. 2, 148–155.
  • A. Bensoussan, Some existence results for stochastic partial differential equations, in Stochastic partial differential equations and applications (Trento, 1990), 37–53, Pritman Res. Notes Math. Ser., 268, Longman Sci. Tech., Harlow, 1992.
  • Ju. A. Dubinskiĭ, Weak convergence for nonlinear elliptic and parabolic equations, Mat. Sb. (N.S.) 67 (109) (1965), 609–642.
  • A. S. Kalashnikov, Some problems of the qualitative theory of second-order nonlinear degenerate parabolic equations, Uspekhi Mat. Nauk 42 (1987), no. 2, 135–176, 287.
  • N. V. Krylov and B. L. Rozovskiĭ, Stochastic evolution equations, in Current problems in mathematics, XIV. (Russian) 256, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Informatsii, Moscow, 1979, pp. 71–147.
  • J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, Gauthier-Villars, Paris, 1969.
  • N. Nagase, Remarks on nonlinear stochastic partial differential equations: an application of the splitting-up method, SIAM J. Control Optim. 33 (1995), no. 6, 1716–1730.
  • E. Pardoux, Equations aux derivées partielles stochastiques non linéaires monotones, Thèse, (1975). (Université Paris XI).
  • P. A. Raviart, Sur la résolution de certaines équations paraboliques non linéaires, J. Functional Analysis 5 (1970), 299–328.
  • M. Sango, On a degenerate quasilinear stochastic parabolic equation, C. R. Math. Acad. Sci. Soc. R. Can. 25 (2003), no. 3, 65–70.
  • J. Simon, Compact sets in the space $L^{p}(0,T;B)$, Ann. Mat. Pura Appl. (4) 146 (1987), 65–96.
  • M. Tsutsumi, On solutions of some doubly nonlinear degenerate parabolic equations with absorption, J. Math. Anal. Appl. 132 (1988), no. 1, 187–212.