Periodic, almost periodic and almost automorphic solutions for SPDEs with monotone coefficients

Abstract

In this paper, we use the variational approach to investigate recurrent properties of solutions for stochastic partial differential equations, which is in contrast to the previous semigroup framework. Consider stochastic differential equations with monotone coefficients. Firstly, we establish the continuous dependence on initial values and coefficients for solutions, which is interesting in its own right. Secondly, we prove the existence of recurrent solutions, which include periodic, almost periodic and almost automorphic solutions. Then we show that these recurrent solutions are globally asymptotically stable in squaremean sense. Finally, for illustration of our results we give two applications, i.e. stochastic reaction diffusion equations and stochastic porous media equations.

Existence, uniqueness and exponential ergodicity under Lyapunov conditions for McKean-Vlasov SDEs with Markovian switching

“…Qiu and Wang [27] get the similar results of recurrent solutions. Cheng and Liu [9] illustrated similar results for SPDEs. In summary, the solutions inherit the convergence of initial values if the coefficients converge point-wise or on any compact set.…”

The second Bogolyubov theorem and global averaging principle for SPDEs with monotone coefficients

“…Proof. Similar to the proof of Proposition 1 in [10], (3.18) can be obtained by Itô's formula, (H6) and Gronwall's lemma.…”

“…Now we discuss the L 2 -bounded solution to equation (3.1) by employing the classical pullback attraction method in random and non-autonomous dynamics (see, e.g. [10,12] etc). For this we need three lemmas.…”

Averaging principle for stochastic complex Ginzburg-Landau equations

“…This result is interesting on its own rights and has been studied extensively; see e.g. [7,9,12,30] and references therein. Without loss of generality, the proof is only given when ε = 1.…”