On a Class of Stochastic Damped Wave Equation

Hatice Taskesen; Beytullah Yağız

doi:10.61383/ejam.20242146

Authors

Hatice Taskesen Department of Mathematics, Faculty of Science, Van Yuzuncu Yil University, Van, Turkey Corresponding Author https://orcid.org/0000-0003-1058-0507
Beytullah Ya˘gız Department of Statistics, Institute of Science, Van Yuzuncu Yil University, Van, Turkey https://orcid.org/0000-0001-5453-185X

DOI:

https://doi.org/10.61383/ejam.20242146

Keywords:

Stochastic wave equation, mild solution, blow up

Abstract

The present work considers a wave equation with multiplicative Gaussian white noise and weak dissipative term on a bounded domain. We first give a theorem including the local existence of mild solutions. An energy bound and a differential inequality are used to give sufficient conditions that provide the blow-up of mild local solutions of the stochastic wave equation. The paper's main contribution comes from handling a multiplicative noise and a general source term contrary to the articles that exist in the literature.

References

[1] Mitsuhiro Nakao, New Trends in the Theory of Hyperbolic Equations, vol. 159, Springer, 2005.

[2] Manoussos G. Grillakis, Regularity and Asymptotic Behavior of the Wave Equation with a Critical Nonlinearity, The Annals of Mathematics 132 (1990), no. 3, 485, Doi 10.2307/1971427.

[3] Kosuke Ono, Blowing-Up and Global Existence of Solutions for some Degenerate Nonlinear Wave Equations with some Dissipation, Nonlinear Analysis: Theory, Methods & Applications 30 (1997), no. 7, 4449–4457.

[4] Salim A Messaoudi, Ala A Talahmeh, and Jamal H Al-Smail, Nonlinear damped wave equation: Existence and blow-up, Computers and Mathematics with Applications 74 (2017), no. 12, 3024–3041, Doi 10.1016/j.camwa.2017.07.048.

[5] Vittorino Pata and Sergey Zelik, Smooth attractors for strongly damped wave equations, Nonlinearity 19 (2006), no. 7, 1495–1506, Doi 10.1088/0951-7715/19/7/001.

[6] S¸evket G¨ ur andMesudeElifUysal, Continuousdependenceofsolutions to the strongly damped nonlinear Klein-Gordon equation, Turkish Journal of Mathematics 42 (2018), no. 3, 904–910, Doi 10.3906/mat-1706-30.

[7] XuRunzhang,Globalexistence, blow up and asymptotic behaviour of solutions for nonlinear Klein-Gordon equation with dissipative term, Mathematical Methods in the Applied Sciences 33 (2010), no. 7, 831–844, Doi 10.1002/mma.1196.

[8] Yanjin Wang, A Sufficient Condition for Finite Time Blow up of the Nonlinear Klein-Gordon Equations with Arbitrarily Positive Initial Energy, Proceedings of the American Mathematical Society 136 (2008), no. 10, 3477–3482.

[9] Ying Wang, Global solutions for a class of nonlinear sixth-order wave equation, Bulletin of the Korean Mathematical Society 55 (2018), no. 4, 1161–1178, Doi 10.4134/BKMS.b170634.

[10] Tae Gab Ha and Jong Yeoul Park, Global existence and uniform decay of a damped KleinGordon equation in a noncylindrical domain, Nonlinear Analysis, Theory, Methods and Applications 74 (2011), no. 2, 577–584, Doi 10.1016/j.na.2010.09.011.

[11] L. Aloui, S. Ibrahim, and K. Nakanishi, Exponential energy decay for damped klein-gordon equation with nonlinearities of arbitrary growth, Communications in Partial Differential Equations 36 (2010), no. 5, 797–818, Doi 10.1080/03605302.2010.534684.

[12] Wenhui Chen, Interplay effects on blow-up of weakly coupled systems for semilinear wave equations with general nonlinear memory terms, Nonlinear Analysis, Theory, Methods and Applications 202 (2021), 112160, Doi 10.1016/j.na.2020.112160.

[13] Pao Liu Chow, Nonlinear stochastic wave equations: Blow-up of second moments in L 2-norm, Annals of Applied Probability 19 (2009), no. 6, 2039–2046, Doi 10.1214/09-AAP602.

[14] Pao Liu Chow, Asymptotics of solutions to semilinear stochastic wave equations, Annals of Applied Probability 16 (2006), no. 2, 757–789, Doi 10.1214/105051606000000141.

[15] Jong Uhn Kim, Periodic and invariant measures for stochastic wave equations, Electronic Journal of Differential Equations 2004 (2004), no. 05, 1–30.

[16] Rana D Parshad, Matthew Beauregard, Aslan Kasimov, Matthew A Beauregard, and Belkacem Said-Houari, Global existence and finite time blow-up in a class of stochastic nonlinear wave equations, Communications on Stochastic Analysis 8 (2014), no. 3, 381–411.

[17] Viorel Barbu and Giuseppe Da Prato, The Stochastic Nonlinear Damped Wave Equation, Appl Math Optim 46 (2002), 125–141, Doi 10.1007/s00245-002-0744-4.

[18] Szymon Peszat and Jerzy Zabczyk, Nonlinear stochastic wave and heat equations, Probability Theory and Related Fields 116 (2000), no. 3, 421–443, Doi 10.1007/s004400050257.

[19] Martin Ondrej´at, Stochastic nonlinear wave equations in local Sobolev spaces, Electronic Journal of Probability 15 (2010), 1041–1091, Doi 10.1214/EJP.v15-789.

[20] Zdzisław Brze´zniak, Martin Ondrej´at, and Jan Seidler, Invariant measures for stochastic nonlinear beam and wave equations, Journal of Differential Equations 260 (2016), no. 5, 4157–4179, Doi 10.1016/j.jde.2015.11.007.

[21] Lijun Bo, Dan Tang, and Yongjin Wang, Explosive solutions of stochastic wave equations with damping on Rd, Journal of Differential Equations 244 (2008), no. 1, 170–187, Doi 10.1016/j.jde.2007.10.016.

[22] Hatice Taskesen, Qualitative results for a relativistic wave equation with multiplicative noise and damping terms, AIMS Mathematics 8 (2023), no. 7, 15232–15254, Doi 10.3934/math.2023778.

[23] Pao Liu Chow, Stochastic wave equations with polynomial nonlinearity, Annals of Applied Probability 12 (2002), no. 1, 361–381, Doi 10.1214/aoap/1015961168.

[24] Guiseppe Da Prato and Jerzy Zabczyk, Stochastic Equations in Infinite Dimensions, second ed., Cambridge University Press, 2014.

[25] R.A. Adams and J.J.F Fournier, Sobolev Spaces, second ed., 2002.

[26] Meng Rong Li and Long Yi Tsai, Existence and nonexistence of global solutions of some system of semilinear wave equations, Nonlinear Analysis, Theory, Methods and Applications 54 (2003), no. 8, 1397–1415, Doi 10.1016/S0362546X(03)00192-5.