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Volume 32, Issue 4
Dynamics of a Non-Linear Stochastic Viscoelastic Equation with Multiplicative Noise

Tomás Caraballo, Nicolás Piña & Jaime Muñoz

DOI: 10.4208/jpde.v32.n4.2

J. Part. Diff. Eq., 32 (2019), pp. 304-325.

Published online: 2020-01

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  • Abstract

The well-posedness and stability properties of a stochastic viscoelastic equation with multiplicative noise, Lipschitz and locally Lipschitz nonlinear terms are investigated. The method of Lyapunov functions is used to investigate the asymptotic dynamics when zero is not a solution of the equation by using an appropriate cocycle and random dynamical system. The stability of mild solutions is proved in both cases of Lipschitz and locally Lipschitz nonlinear terms. Furthermore, we investigate the existence of a non-trivial stationary solution which is exponentially stable, by using a general random fixed point theorem for general cocycles. In this case, the stationary solution is generated by the composition of random variable and Wiener shift. In addition, the theory of random dynamical system is used to construct another cocycle and prove the existence of a random fixed point exponentially attracting every path.

  • Keywords

Stochastic viscoelastic exponential stability stabilization random dynamical systems attractors.

  • AMS Subject Headings

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

caraball@us.es (Tomás Caraballo)

jose.pina1301@alumnos.ubiobio.cl (Nicolás Piña)

jedmunoz@ubiobio.cl (Jaime Muñoz)

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  • TXT
@Article{JPDE-32-304, author = {Caraballo , TomásPiña , Nicolás and Muñoz , Jaime}, title = {Dynamics of a Non-Linear Stochastic Viscoelastic Equation with Multiplicative Noise}, journal = {Journal of Partial Differential Equations}, year = {2020}, volume = {32}, number = {4}, pages = {304--325}, abstract = {

The well-posedness and stability properties of a stochastic viscoelastic equation with multiplicative noise, Lipschitz and locally Lipschitz nonlinear terms are investigated. The method of Lyapunov functions is used to investigate the asymptotic dynamics when zero is not a solution of the equation by using an appropriate cocycle and random dynamical system. The stability of mild solutions is proved in both cases of Lipschitz and locally Lipschitz nonlinear terms. Furthermore, we investigate the existence of a non-trivial stationary solution which is exponentially stable, by using a general random fixed point theorem for general cocycles. In this case, the stationary solution is generated by the composition of random variable and Wiener shift. In addition, the theory of random dynamical system is used to construct another cocycle and prove the existence of a random fixed point exponentially attracting every path.

}, issn = {2079-732X}, doi = {https://doi.org/10.4208/jpde.v32.n4.2}, url = {http://global-sci.org/intro/article_detail/jpde/13611.html} }
TY - JOUR T1 - Dynamics of a Non-Linear Stochastic Viscoelastic Equation with Multiplicative Noise AU - Caraballo , Tomás AU - Piña , Nicolás AU - Muñoz , Jaime JO - Journal of Partial Differential Equations VL - 4 SP - 304 EP - 325 PY - 2020 DA - 2020/01 SN - 32 DO - http://doi.org/10.4208/jpde.v32.n4.2 UR - https://global-sci.org/intro/article_detail/jpde/13611.html KW - Stochastic viscoelastic KW - exponential stability KW - stabilization KW - random dynamical systems KW - attractors. AB -

The well-posedness and stability properties of a stochastic viscoelastic equation with multiplicative noise, Lipschitz and locally Lipschitz nonlinear terms are investigated. The method of Lyapunov functions is used to investigate the asymptotic dynamics when zero is not a solution of the equation by using an appropriate cocycle and random dynamical system. The stability of mild solutions is proved in both cases of Lipschitz and locally Lipschitz nonlinear terms. Furthermore, we investigate the existence of a non-trivial stationary solution which is exponentially stable, by using a general random fixed point theorem for general cocycles. In this case, the stationary solution is generated by the composition of random variable and Wiener shift. In addition, the theory of random dynamical system is used to construct another cocycle and prove the existence of a random fixed point exponentially attracting every path.

Caraballo , TomásPiña , Nicolás and Muñoz , Jaime. (2020). Dynamics of a Non-Linear Stochastic Viscoelastic Equation with Multiplicative Noise. Journal of Partial Differential Equations. 32 (4). 304-325. doi:10.4208/jpde.v32.n4.2
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