Localization of solutions to stochastic porous media equations: finite speed of propagation

Viorel Barbu (Al.I.Cuza University)
Michael Roeckner (Bielefeld University)


It is proved that the solutions to the slow diffusion stochastic porous media equation $dX-{\Delta}( |X|^{m-1}X )dt=\sigma(X)dW_t,$ $ 1< m\le 5,$ in $\mathcal{O}\subset\mathbb{R}^d,\ d=1,2,3,$ have the property of finite speed of propagation of disturbances for $\mathbb{P}\text{-a.s.}$ ${\omega}\in{\Omega}$ on a sufficiently small time interval $(0,t({\omega}))$.

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Pages: 1-11

Publication Date: January 29, 2012

DOI: 10.1214/EJP.v17-1768


  • Antontsev, S. N. On the localization of solutions of nonlinear degenerate elliptic and parabolic equations. (Russian) Dokl. Akad. Nauk SSSR 260 (1981), no. 6, 1289--1293. MR0636152
  • S.N. Antontsev, J.I. Diaz, On space or time localization of solutions of nonlinear elliptic or parabolic equations via energy methods, in "Recent Advances in nonlinear Elliptic and Parabolic Problems", Ph. Benilan et al. (eds.), Pitman Research Notes in Mathematics, Longman, 1983, 2-14.
  • Antontsev, S. N.; Díaz, J. I.; Shmarev, S. Energy methods for free boundary problems. Applications to nonlinear PDEs and fluid mechanics. Progress in Nonlinear Differential Equations and their Applications, 48. Birkhäuser Boston, Inc., Boston, MA, 2002. xii+329 pp. ISBN: 0-8176-4123-8 MR1858749
  • Antontsev, S. N.; Shmarev, S. I. A model porous medium equation with variable exponent of nonlinearity: existence, uniqueness and localization properties of solutions. Nonlinear Anal. 60 (2005), no. 3, 515--545. MR2103951
  • Barbu, Viorel; Da Prato, Giuseppe; Röckner, Michael. Existence and uniqueness of nonnegative solutions to the stochastic porous media equation. Indiana Univ. Math. J. 57 (2008), no. 1, 187--211. MR2400255
  • Barbu, Viorel; Da Prato, Giuseppe; Röckner, Michael. Stochastic porous media equations and self-organized criticality. Comm. Math. Phys. 285 (2009), no. 3, 901--923. MR2470909
  • Barbu, Viorel; Da Prato, Giuseppe; Röckner, Michael. Finite time extinction for solutions to fast diffusion stochastic porous media equations. C. R. Math. Acad. Sci. Paris 347 (2009), no. 1-2, 81--84. MR2536755
  • Barbu, Viorel; Da Prato, Giuseppe. Internal stabilization by noise of the Navier-Stokes equation. SIAM J. Control Optim. 49 (2011), no. 1, 1--20. MR2765654
  • Barbu, Viorel; Röckner, Michael. On a random scaled porous media equation. J. Differential Equations 251 (2011), no. 9, 2494--2514. MR2825337
  • Da Prato, Giuseppe. Kolmogorov equations for stochastic PDEs. Advanced Courses in Mathematics. CRM Barcelona. Birkhäuser Verlag, Basel, 2004. x+182 pp. ISBN: 3-7643-7216-8 MR2111320
  • Díaz, J. I. Qualitative study of nonlinear parabolic equations: an introduction. Extracta Math. 16 (2001), no. 3, 303--341. MR1897752
  • Díaz, J. Ildefonso; Véron, Laurent. Local vanishing properties of solutions of elliptic and parabolic quasilinear equations. Trans. Amer. Math. Soc. 290 (1985), no. 2, 787--814. MR0792828
  • P. Gess, Strong solutions for stochastic partial differential equations of gradient type, Preprint 2011, arXiv:1104.4243
  • Lototsky, S. V. A random change of variables and applications to the stochastic porous medium equation with multiplicative time noise. Commun. Stoch. Anal. 1 (2007), no. 3, 343--355. MR2403855
  • Prévôt, Claudia; Röckner, Michael. A concise course on stochastic partial differential equations. Lecture Notes in Mathematics, 1905. Springer, Berlin, 2007. vi+144 pp. ISBN: 978-3-540-70780-6; 3-540-70780-8 MR2329435
  • Röckner, Michael; Wang, Feng-Yu. Non-monotone stochastic generalized porous media equations. J. Differential Equations 245 (2008), no. 12, 3898--3935. MR2462709
  • Ren, Jiagang; Röckner, Michael; Wang, Feng-Yu. Stochastic generalized porous media and fast diffusion equations. J. Differential Equations 238 (2007), no. 1, 118--152. MR2334594
  • Vázquez, Juan Luis. The porous medium equation. Mathematical theory. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford, 2007. xxii+624 pp. ISBN: 978-0-19-856903-9; 0-19-856903-3 MR2286292

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