Asymptotics of the mild solution of awave equation in three-dimensional space driven by a general stochastic measure
DOI:
https://doi.org/10.17721/1812-5409.2019/4.2Abstract
We study the Cauchy problem for a wave equation in three-dimensional space driven by a general stochastic measure. Under some assumptions, we prove that the mild solution tends to zero almost surely as the absolute value of the spatial variable tends to infinity.
Key words: asymptotic behavior, stochastic measure, stochastic wave equation, mild solution, Besov space.
Pages of the article in the issue: 12 - 17
Language of the article: English
References
KWAPIE´N, S.,WOYCZY´NSKI, W.A. (1992) Random Series and Stochastic Integrals: Single and Multiple. Boston: Birkh¨auser, 360 p.
MILLET, A., MORIEN, P.-L. (2000) On a stochastic wave equation in two space dimensions: regularity of the solution and its density. Stochastic Processes and their Applications. 86, p. 141–162.
RADCHENKO, V.N. (1997) On a definition of the integral of a random function, Theory of Probability and its Applications. 41, no. 3, p. 597–601.
BODNARCHUK, I.M. (2017) Wave equation with a stochastic measure. Theory of Probability and Mathematical Statistics. 94, p. 1–16.
BODNARCHUK, I.M., RADCHENKO, V.M. (2019) Wave equation in the plane driven by a general stochastic measure. Theory of Probability and Mathematical Statistics. 98, p. 73–90.
BODNARCHUK, I.M., RADCHENKO, V.M. (2019) Wave equation in three-dimensional space driven by a general stochastic measure, Teoriya Imovirnostei ta Matematychna Statystyka. 1(100), p. 43–59. (In Ukrainian)
BODNARCHUK, I.M., RADCHENKO, V.M. (2012) Asymptotic behavior of a solution of the heat equation with general stochastic measure. Scientific Herald of Yuriy Fedkovych Chemivtsi National University. Series: mathematics. 2, № 1, p. 7–11. (In Ukrainian)
BODNARCHUK, I. (2018) Asymptotics of the mild solution for the wave equation with a stochastic measure. Vesnik of Brest University. Series 4. Physics. Mathematics. № 2, p. 70–78. (In Russian)
BODNARCHUK, I. (2016) Asymptotic behavior of a mild solution of the stochastic heat equation. Bulletin of Taras Shevchenko National University of Kyiv. Maths. Mechanics. № (2)36, p. 40–42. (In Ukrainian)
RADCHENKO, V.M. (2012) Asymptotic behavior of a solution for the heat equation with a stochastic measure as t → ∞. Scientific Bulletin of Uzhhorod University. Series of Mathematics and Informatics. Iss. 23, № 1, p. 119–124. (In Ukrainian)
YANG, M., DUAN, J., KLOEDEN, P. (2011) Asymptotic behavior of solutions for random wave equations with nonlinear damping and white noise. Nonlinear Analysis: Real World Applications. Vol. 12, no 1, p. 464–478.
CHEN, L., DALANG, R.C. (2015) Moment bounds and asymptotics for the stochastic wave equation. Stochastic Processes and their Applications. Vol. 125, p. 1605–1628.
JIANG, Y., WANG, S., WANG, X. (2019) Asymptotics of the Solutions to Stochastic Wave Equations Driven by a Non-Gaussian L´evy Process. Acta Mathematica Scientia. 39, p. 731–746.
VLADIMIROV, V.S. (1981) Equations of mathematical physics. Moscow: Nauka, 512 p. (In Russian)
RADCHENKO, V. (2009) Mild solution of the heat equation with a general stochastic measure. Studia Mathematica. 194, no. 3, p. 231–251.
Downloads
Published
Issue
Section
License
Authors who publish with this journal agree to the following terms:
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).
