In this paper, a stochastic strongly damped wave equation with polynomial drift and diffusion terms is studied. First we prove there exists a unique solution for this equation by using a truncation method. Then, we establish the tightness of a family of probability distributions of solutions and obtain the existence of invariant measures by introducing an appropriate Lyapunov function and utilizing a decomposition approach. Finally, the regularity of the invariant measure is investigated, that is, the invariant measure is supported by a regular space.
REFERENCES
1.
A. N.
Carvalho
and J. W.
Cholewa
, “Local well posedness for strongly damped wave equations with critical nonlinearities
,” Bull. Aust. Math. Soc.
66
(3
), 443
–463
(2002
).2.
J.
Shatah
and M.
Struwe
, “Well-posedness in the energy space for semilinear wave equations with critical growth
,” Int. Math. Res. Not.
7
, 303
–309
(1994
).3.
Z.
Yang
and P.
Ding
, “Longtime dynamics of Kirchhoff equation with strong damping and critical nonlinearity on
,” J. Math. Anal. Appl.
434
(2
), 1826
–1851
(2016
).4.
S.
Zhou
, “Attractors for strongly damped wave equations with critical exponent
,” Appl. Math. Lett.
16
(8
), 1307
–1314
(2003
).5.
P.
Chow
, “Stochastic wave equations with polynomial nonlinearity
,” Ann. Appl. Probab.
12
(1
), 361
–381
(2002
).6.
Y.
Li
, B.
Li
, and X.
Li
, “Uniform random attractors for a non-autonomous stochastic strongly damped wave equation on
,” Z. Angew. Math. Phys.
73
(3
), 106
(2022
).7.
M.
Ondreját
, “Existence of global mild and strong solutions to stochastic hyperbolic evolution equations driven by a spatially homogeneous Wiener process
,” J. Evol. Equations
4
(2
), 169
–191
(2004
).8.
S.
Peszat
, “The Cauchy problem for a nonlinear stochastic wave equation in any dimension
,” J. Evol. Equations
2
(3
), 383
–394
(2002
).9.
B.
Wang
, “Well-posedness and long term behavior of supercritical wave equations driven by nonlinear colored noise on
,” J. Funct. Anal.
283
(2
), 109498
(2022
).10.
Z.
Wang
and S.
Zhou
, “Existence and upper semicontinuity of random attractors for non-autonomous stochastic strongly damped wave equation with multiplicative noise
,” Discrete Contin. Dyn. Syst. Ser. A
37
(5
), 2787
–2812
(2017
).11.
Z.
Wang
and L.
Zhang
, “Finite fractal dimension of random attractor for stochastic non-autonomous strongly damped wave equation
,” Comput. Math. Appl.
75
(9
), 3343
–3357
(2018
).12.
N.
Krylov
and N.
Bogolyubov
, “La théorie générale de la mesure dans son application à l’étude des systémes de la Mécanique non linéaire
,” Ann. Math
38
, 65
–113
(1937
).13.
Z.
Chen
and B.
Wang
, “Limit measures and ergodicity of fractional stochastic reaction–diffusion equations on unbounded domains
,” Stochastics Dyn.
22
(2
), 2140012
(2022
).14.
G.
Da Prato
and J.
Zabczyk
, Stochastic Equations in Infinite Dimensions
, 2nd ed. (Cambridge University Press
, Cambridge
, 1992
).15.
Y.
Lin
, Y.
Li
, and D.
Li
, “Periodic measures of impulsive stochastic neural networks lattice systems with delays
,” J. Math. Phys.
63
(12
), 122702
(2022
).16.
D.
Li
, B.
Wang
, and X.
Wang
, “Periodic measures of stochastic delay lattice systems
,” J. Differ. Equations
272
, 74
–104
(2021
).17.
B.
Wang
, “Dynamics of fractional stochastic reaction-diffusion equations on unbounded domains driven by nonlinear noise
,” J. Differ. Equations
268
(1
), 1
–59
(2019
).18.
H.
Bessaih
and B.
Ferrario
, “Statistical properties of stochastic 2D Navier-Stokes equations from linear models
,” Discrete Contin. Dyn. Syst., Ser. B
21
(9
), 2927
–2947
(2016
).19.
V.
Barbu
and G. D.
Prato
, “The stochastic nonlinear damped wave equation
,” Appl. Math. Optim.
46
(2
), 125
–141
(2002
).20.
K. R.
Parthasarathy
, Probability Measures on Metric Spaces
(Academic Press
, New York; London
, 1967
).21.
J. U.
Kim
, “Periodic and invariant measures for stochastic wave equations
,” Electron. J. Differ. Equations
5
, 30
(2004
).22.
J. U.
Kim
, “On the stochastic wave equation with nonlinear damping
,” Appl. Math. Optim.
58
(1
), 29
–67
(2008
).23.
Z.
Brzeźniak
, M.
Ondreját
, and J.
Seidler
, “Invariant measures for stochastic nonlinear beam and wave equations
,” J. Differ. Equations
260
(5
), 4157
–4179
(2016
).24.
V.
Barbu
, G.
Da Prato
, and L.
Tubaro
, “Stochastic wave equations with dissipative damping
,” Stochastic Processes Appl.
117
(8
), 1001
–1013
(2007
).25.
Y.
Jiang
, X.
Wang
, and Y.
Wang
, “Stochastic wave equation of pure jumps: Existence, uniqueness and invariant measures
,” Nonlinear Anal.: Theory, Methods Appl.
75
(13
), 5123
–5138
(2012
).26.
R.
Wang
and Y.
Li
, “Asymptotic behavior of stochastic discrete wave equations with nonlinear noise and damping
,” J. Math. Phys.
61
(5
), 052701
(2020
).27.
Z.
Yang
, P.
Ding
, and L.
Li
, “Longtime dynamics of the Kirchhoff equations with fractional damping and supercritical nonlinearity
,” J. Math. Anal. Appl.
442
(2
), 485
–510
(2016
).28.
C.
Prevot
and M.
Rockner
, “A concise course on stochastic partial differential equations
,” in Lecture Notes in Mathematics
(Springer
, Berlin
, 2007
).29.
S. S.
Sritharan
and P.
Sundar
, “Large deviations for the two-dimensional Navier–Stokes equations with multiplicative noise
,” Stochastic Processes Appl.
116
(11
), 1636
–1659
(2006
).© 2025 Author(s). Published under an exclusive license by AIP Publishing.
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