Exponential behavior and upper noise excitation index of solutions to evolution equations with unbounded delay and tempered fractional Brownian motions


In this paper, we investigate stochastic evolution equations with unbounded delay in fractional power spaces perturbed by a tempered fractional Brownian motion \(B_Q^{\sigma ,\lambda }(t)\) with \(-1/2<\sigma <0\) and \(\lambda >0\). We first introduce a technical lemma which is crucial in our stability analysis. Then, we prove the existence and uniqueness of mild solutions by using semigroup methods. The upper nonlinear noise excitation index of the energy solutions at any finite time t is also obtained. Finally, we consider the exponential asymptotic behavior of mild solutions in mean square.

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Correspondence to Tomás Caraballo.

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This work was supported by NSF of China (Grant No. 41875084), the Fundamental Research Funds for the Central Universities under Grant Nos. lzujbky-2018-ot03 and lzujbky-2018-it58. The research of T. Caraballo has been partially supported by Ministerio de Ciencia Innovación y Universidades (Spain), FEDER (European Community) under Grant PGC2018-096540-B-I00, and by Junta de Andalucía (Consejería de Economía y Conocimiento) and FEDER under Projects US-1254251 and P18-FR-4509.

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Wang, Y., Liu, Y. & Caraballo, T. Exponential behavior and upper noise excitation index of solutions to evolution equations with unbounded delay and tempered fractional Brownian motions. J. Evol. Equ. (2021). https://doi.org/10.1007/s00028-020-00656-0

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  • Stochastic PDEs
  • Unbounded delay
  • Tempered fractional Brownian motion
  • Fractional powers of closed operators
  • Exponential decay in mean square