Asymptotic behaviour of a stochastic semilinear dissipative
functional equation without uniqueness of solutions

Caraballo, Tomás; Garrido–Atienza, María J.; Schmalfuß, Björn; Valero, José

doi:10.3934/dcdsb.2010.14.439

Cited by 100 publications

(78 citation statements)

References 13 publications

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“…It is shown in Ref. [7] that N is a Polish space for any N . Since z(θ t ω) = m j =1 h j z j (θ t ω j ) and …”

Section: The Measurability Of the Pullback Attractorsmentioning

confidence: 98%

“…From Ref. [7], we know that P N is a finite measure on ( N , F N ), and that if A ∈F N , then A ∈F . Proof.…”

Section: Lemma 52 the Map R × N (T ω) → Z(θ T ω) Is Continuous In mentioning

confidence: 99%

“…Some ideas for checking the asymptotical upper-semicompactness of solutions in L p (R n ) and H 1 (R n ) are developed in this paper to circumvent the difficulty caused by the nonlinearity and the multi-valued case. The reader is also referred to [5][6][7]9,15,17,18,22,27] for asymptotic behavior of multi-valued dynamical systems.…”

Section: Introductionmentioning

confidence: 99%

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Pullback attractors for multi-valued non-compact random dynamical systems generated by reaction–diffusion equations on an unbounded domain

Wang

2015

Journal of Differential Equations

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In this paper we study pullback attractors of reaction-diffusion equations on an unbounded domain with non-autonomous deterministic as well as stochastic forcing terms for which the uniqueness of solutions need not hold. We first present the existence and structure of pullback attractors of multi-valued non-compact random dynamical systems. Then we prove the existence of pullback attractors in L 2 (R n ), L p (R n ) and H 1 (R n ) for multi-valued non-compact random dynamical systems associated with the reaction-diffusion equations on R n , and the identical relation of pullback attractors in different spaces is also provided. In particular, the measurability of pullback attractors is established for reaction-diffusion equations on R n .

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“…It is shown in Ref. [7] that N is a Polish space for any N . Since z(θ t ω) = m j =1 h j z j (θ t ω j ) and …”

Section: The Measurability Of the Pullback Attractorsmentioning

confidence: 98%

“…From Ref. [7], we know that P N is a finite measure on ( N , F N ), and that if A ∈F N , then A ∈F . Proof.…”

Section: Lemma 52 the Map R × N (T ω) → Z(θ T ω) Is Continuous In mentioning

confidence: 99%

See 1 more Smart Citation

Pullback attractors for multi-valued non-compact random dynamical systems generated by reaction–diffusion equations on an unbounded domain

Wang

2015

Journal of Differential Equations

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“…Then it is not difficult to check that Ω = ∪ N Ω N and Ω N ∈ F. It is shown in [7] that Ω N is a polish space for any N . We need the continuity of the map R× Ω N ∋ (t, ω) → z (θ t ω).…”

Section: The Random Attractormentioning

confidence: 99%

“…Also, letF ΩN be the completion of F ΩN with respect to P ΩN . The following facts are proved in [7]:…”

Section: The Random Attractormentioning

confidence: 99%

Random attractors for stochastic lattice systems with non-Lipschitz nonlinearity

Caraballo¹,

Morillas²,

Valero³

2011

Journal of Difference Equations and Applications

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In this paper we study the asymptotic behaviour of solutions of a first-order stochastic lattice dynamical system with an additive noise.We do not assume any Lipschitz condition on the nonlinear term, just a continuity assumption together with growth and dissipative conditions, so that uniqueness of the Cauchy problem fails to be true.Using the theory of multi-valued random dynamical systems we prove the existence of a random compact global attractor.

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Some Aspects Concerning the Dynamics of Stochastic Chemostats

Caraballo

Garrido–Atienza

López‐de‐la‐Cruz

2016

Advances in Dynamical Systems and Control

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In this paper we study a simple chemostat model influenced by white noise which makes this kind of models more realistic. We use the theory of random attractors and, to that end, we first perform a change of variable using the OrnsteinUhlenbeck process, transforming our stochastic model into a system of differential equations with random coefficients. After proving that this random system possesses a unique solution for any initial value, we analyze the existence of random attractors. Finally we illustrate our results with some numerical simulations.

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Asymptotic behaviour of a stochastic semilinear dissipative functional equation without uniqueness of solutions

Cited by 100 publications

References 13 publications

Pullback attractors for multi-valued non-compact random dynamical systems generated by reaction–diffusion equations on an unbounded domain

Pullback attractors for multi-valued non-compact random dynamical systems generated by reaction–diffusion equations on an unbounded domain

Random attractors for stochastic lattice systems with non-Lipschitz nonlinearity

Some Aspects Concerning the Dynamics of Stochastic Chemostats

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