
Previous Article
Influence of latent period and nonlinear incidence rate on the dynamics of SIRS epidemiological models
 DCDSB Home
 This Issue

Next Article
Existence of traveling wave solutions for a nonlocal reactiondiffusion model of influenza a drift
Limit behavior of nonlinear stochastic wave equations with singular perturbation
1.  Department of Mathematics, Nanjing University, Nanjing, 210093, China 
2.  School of Science, Nanjing University of Science & Technology, Nanjing, 210094, China 
$ \nu $u_{tt}$+u_t=$Δ$u+f(u)+$ε$\dot{W}$
on an open bounded domain $D\subset\R^n$, $1\leq n\leq 3$,, is studied in the sense of distribution for small $\nu, $ε$>0$. Here $\nu$ is the parameter that describes the singular perturbation. First, by a decomposition of Markov semigroup defined by (1), a stationary solution is constructed which describes the asymptotic behavior of solution from initial value in state space $H_0^1(D)\times L^2(D)$. Then a global measure attractor is constructed for (1). Furthermore under the case that the stochastic force is proportional to the square root of singular perturbation, that is ε$=\sqrt{\nu}$, we study the limit of the behavior of all the stationary solutions of (1) as $\nu\rightarrow 0$. It is shown that, by studying a continuity property on $\nu$ for the measure attractors of (1), any one stationary solution of the limit equation
$u_t=$Δ$u+f(u).$
is a limit point of a stationary solution of (1), as $\nu\rightarrow 0$.
[1] 
Petr Kůrka. On the measure attractor of a cellular automaton. Conference Publications, 2005, 2005 (Special) : 524535. doi: 10.3934/proc.2005.2005.524 
[2] 
Fabio Camilli, Giulia Cavagnari, Raul De Maio, Benedetto Piccoli. Superposition principle and schemes for measure differential equations. Kinetic & Related Models, 2021, 14 (1) : 89113. doi: 10.3934/krm.2020050 
[3] 
Kai Liu. Stationary solutions of neutral stochastic partial differential equations with delays in the highestorder derivatives. Discrete & Continuous Dynamical Systems  B, 2018, 23 (9) : 39153934. doi: 10.3934/dcdsb.2018117 
[4] 
Giuseppe Da Prato. An integral inequality for the invariant measure of some finite dimensional stochastic differential equation. Discrete & Continuous Dynamical Systems  B, 2016, 21 (9) : 30153027. doi: 10.3934/dcdsb.2016085 
[5] 
Zhiyuan Wen, Meirong Zhang. On the optimization problems of the principal eigenvalues of measure differential equations with indefinite measures. Discrete & Continuous Dynamical Systems  B, 2020, 25 (8) : 32573274. doi: 10.3934/dcdsb.2020061 
[6] 
Lorenzo Zambotti. A brief and personal history of stochastic partial differential equations. Discrete & Continuous Dynamical Systems, 2021, 41 (1) : 471487. doi: 10.3934/dcds.2020264 
[7] 
Arnulf Jentzen. Taylor expansions of solutions of stochastic partial differential equations. Discrete & Continuous Dynamical Systems  B, 2010, 14 (2) : 515557. doi: 10.3934/dcdsb.2010.14.515 
[8] 
Xiangfeng Yang. Stability in measure for uncertain heat equations. Discrete & Continuous Dynamical Systems  B, 2019, 24 (12) : 65336540. doi: 10.3934/dcdsb.2019152 
[9] 
Neil S. Trudinger, XuJia Wang. Quasilinear elliptic equations with signed measure. Discrete & Continuous Dynamical Systems, 2009, 23 (1&2) : 477494. doi: 10.3934/dcds.2009.23.477 
[10] 
Fabio Camilli, Raul De Maio. Memory effects in measure transport equations. Kinetic & Related Models, 2019, 12 (6) : 12291245. doi: 10.3934/krm.2019047 
[11] 
Ugo Bessi. The stochastic value function in metric measure spaces. Discrete & Continuous Dynamical Systems, 2017, 37 (4) : 18191839. doi: 10.3934/dcds.2017076 
[12] 
Mieczysław Cichoń, Bianca Satco. On the properties of solutions set for measure driven differential inclusions. Conference Publications, 2015, 2015 (special) : 287296. doi: 10.3934/proc.2015.0287 
[13] 
Elena Goncharova, Maxim Staritsyn. On BVextension of asymptotically constrained controlaffine systems and complementarity problem for measure differential equations. Discrete & Continuous Dynamical Systems  S, 2018, 11 (6) : 10611070. doi: 10.3934/dcdss.2018061 
[14] 
Min Yang, Guanggan Chen. Finite dimensional reducing and smooth approximating for a class of stochastic partial differential equations. Discrete & Continuous Dynamical Systems  B, 2020, 25 (4) : 15651581. doi: 10.3934/dcdsb.2019240 
[15] 
Qi Lü, Xu Zhang. A concise introduction to control theory for stochastic partial differential equations. Mathematical Control & Related Fields, 2021 doi: 10.3934/mcrf.2021020 
[16] 
Tomás Caraballo, José Real, T. Taniguchi. The exponential stability of neutral stochastic delay partial differential equations. Discrete & Continuous Dynamical Systems, 2007, 18 (2&3) : 295313. doi: 10.3934/dcds.2007.18.295 
[17] 
Sergio Albeverio, Sonia Mazzucchi. Infinite dimensional integrals and partial differential equations for stochastic and quantum phenomena. Journal of Geometric Mechanics, 2019, 11 (2) : 123137. doi: 10.3934/jgm.2019006 
[18] 
Tomasz Kosmala, Markus Riedle. Variational solutions of stochastic partial differential equations with cylindrical Lévy noise. Discrete & Continuous Dynamical Systems  B, 2021, 26 (6) : 28792898. doi: 10.3934/dcdsb.2020209 
[19] 
Zhongkai Guo. Invariant foliations for stochastic partial differential equations with dynamic boundary conditions. Discrete & Continuous Dynamical Systems, 2015, 35 (11) : 52035219. doi: 10.3934/dcds.2015.35.5203 
[20] 
Mogtaba Mohammed, Mamadou Sango. Homogenization of nonlinear hyperbolic stochastic partial differential equations with nonlinear damping and forcing. Networks & Heterogeneous Media, 2019, 14 (2) : 341369. doi: 10.3934/nhm.2019014 
2020 Impact Factor: 1.327
Tools
Metrics
Other articles
by authors
[Back to Top]