The dynamics of the mean mass of a solution of the stochastic porous media equation
Author:
S. A. Mel’nik
Translated by:
N. Semenov
Journal:
Theor. Probability and Math. Statist. 87 (2013), 153-161
MSC (2010):
Primary 60F10; Secondary 62F05
DOI:
https://doi.org/10.1090/S0094-9000-2014-00910-2
Published electronically:
March 21, 2014
MathSciNet review:
3241452
Full-text PDF Free Access
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Additional Information
Abstract: Conditions are found under which the mean mass of a non-trivial solution of the Cauchy problem for the stochastic porous media equation becomes infinitely large during a finite time.
References
- Viorel Barbu, Giuseppe Da Prato, and Michael Röckner, Existence of strong solutions for stochastic porous media equation under general monotonicity conditions, Ann. Probab. 37 (2009), no. 2, 428–452. MR 2510012, DOI https://doi.org/10.1214/08-AOP408
- Xinfeng Liu and Mingxin Wang, The critical exponent of doubly singular parabolic equations, J. Math. Anal. Appl. 257 (2001), no. 1, 170–188. MR 1824673, DOI https://doi.org/10.1006/jmaa.2000.7341
- S. A. Mel′nik, The dynamics of solutions of the Cauchy problem for a stochastic equation of parabolic type with power non-linearities (the stochastic term is linear), Teor. Ĭmovīr. Mat. Stat. 64 (2001), 110–117 (Ukrainian, with Ukrainian summary); English transl., Theory Probab. Math. Statist. 64 (2002), 129–137. MR 1922959
- Nobuyuki Ikeda and Shinzo Watanabe, Stochastic differential equations and diffusion processes, 2nd ed., North-Holland Mathematical Library, vol. 24, North-Holland Publishing Co., Amsterdam; Kodansha, Ltd., Tokyo, 1989. MR 1011252
- S. A. Mel′nik, The existence of solutions of parabolic-type stochastic equations with power nonlinearities, Teor. Ĭmovīr. Mat. Stat. 53 (1995), 103–108 (Ukrainian, with Ukrainian summary); English transl., Theory Probab. Math. Statist. 53 (1996), 113–118 (1997). MR 1449111
- S. A. Mel’nik, Maximum principle for stochastic combustion equations, Dep. UKRINTEI, 434-Uk 921 06.04.92, 1992, 1–5. (Russian)
- Claudia Prévôt and Michael Röckner, A concise course on stochastic partial differential equations, Lecture Notes in Mathematics, vol. 1905, Springer, Berlin, 2007. MR 2329435
References
- V. Barbu, G. Da Prato, and M. Röckner, Existence of strong solutions of stochastic porous media equations, Ann. Probab. 37 (2009), 428–452. MR 2510012 (2011a:35592)
- X. Liu and M. Wang, The critical exponent of doubly singular parabolic equations, J. Math. Anal. Appl. 257 (2001), 170–188. MR 1824673 (2002c:35141)
- S. A. Mel’nik, The dynamics of solutions of the Cauchy problem for a stochastic equation of parabolic type with power non-linearities (the stochastic term is linear), Teor. Imovir. Mat. Stat. 64 (2001), 110–117; English transl. in Theory Probab. Math. Statist. 64 (2002), 129–137. MR 1922959 (2003m:60056)
- N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, second edition, North-Holland Publishing Co., Amsterdam–Kodansha, Ltd., Tokyo, 1989. MR 1011252 (90m:60069)
- S. A. Mel’nik, The existence of solutions of parabolic-type stochastic equations with power non-linearities, Teor. Imovir. Mat. Stat. 53 (1995), 103–108; English transl. in Theory Probab. Math. Statist. 53 (1996), 113–118. MR 1449111 (98c:60069)
- S. A. Mel’nik, Maximum principle for stochastic combustion equations, Dep. UKRINTEI, 434-Uk 921 06.04.92, 1992, 1–5. (Russian)
- C. Prevot and M. Röckner, A Concise Course on Stochastic Partial Differential Equations, Springer-Verlag, Berlin, 2007. MR 2329435 (2009a:60069)
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Additional Information
S. A. Mel’nik
Affiliation:
Department of Probability Theory and Mathematical Statistics, Institute of Applied Mathematics and Mechanics, National Academy of Science of Ukraine, R. Luxemburg Street, 74, Donetsk, 83114, Ukraine
Email:
melnik@iamm.ac.donetsk.ua
Keywords:
Stochastic partial differential equation
Received by editor(s):
December 13, 2011
Published electronically:
March 21, 2014
Additional Notes:
This research was supported by the State Fund for Fundamental Researches of Ukraine and Russian Fund for Fundamental Researches, grant $\Phi$40.1/023
Article copyright:
© Copyright 2014
American Mathematical Society
