Averaging Principles for Nonautonomous Two-Time-Scale Stochastic Reaction-Diffusion Equations with Jump

Abstract

In this paper, we aim to develop the averaging principle for a slow-fast system of stochastic reaction-diffusion equations driven by Poisson random measures. The coefficients of the equation are assumed to be functions of time, and some of them are periodic or almost periodic. Therefore, the Poisson term needs to be processed, and a new averaged equation needs to be given. For this reason, the existence of time-dependent evolution family of measures associated with the fast equation is studied and proved that it is almost periodic. Next, according to the characteristics of almost periodic functions, the averaged coefficient is defined by the evolution family of measures, and the averaged equation is given. Finally, the validity of the averaging principle is verified by using the Khasminskii method.

Strong averaging principles for a class of non-autonomous slow-fast systems of SPDEs with polynomial growth

“…Using the same argument as Lemma 4.1 in [22], we also can get that there exists δ > 0, such that for any p ≥ 1, have…”

“…Finally, for i = 1, 2, denote the realization of the operators A i and L in E are A i and L, and the operator A i generates an analytic semigroup e tA i . Now, we give the following assumptions about the operators A i and Q i as in [20] and [22].…”

“…Next, by proceeding as Lemma 3.1 in [11] and Lemma 3.1 in [22], we can prove that the solution X ǫ,n (t) and Y ǫ,n (t) of equation (3.6) are bounded, the detailed proof will be provided in the Appendix. Lemma 3.2.…”

“…In 2017, Cerrai [20] studied the averaging principle for nonautonomous slow-fast systems of stochastic reaction-diffusion equations driven by Gaussian noises. Then, Liu [21] researched the averaging principle for non-autonomous stochastic differential equations driven by Gaussian noises and Xu [22] studied the averaging principle for non-autonomous slow-fast systems of stochastic reaction-diffusion equations driven by Gaussian noises and Poisson random measures.…”

“…For example, one of the reaction-diffusion equations is the Fitzhugh-Nagumo or Ginzburg-Landau type, in which the coefficients of these systems satisfy the polynomial growth condition, has appeared in the fields of biology and physics and attracted considerable attention. As a consequence, comparing with the work of Xu [22], the coefficients of the original equation (1.1) in this work are extended to satisfy the polynomial growth condition and the local Lipschitz condition. In order to deal with it, subdifferential is considered to obtain some a-priori bounds and the stopping technique will be used very frequently.…”