Tomás Caraballo

Under the guidelines of probability intuitions and stochastic perspectives of tempered Lévy flights, a class of tempered fractional nonlocal lattice systems is introduced for the first time which can be regarded as the generalized form of tempered fractional p-Laplacian system. Correspondingly, some property of discrete Orlicz spaces is established. Based on this framework, we first show the global existence of solutions for the lattice system via an existence theorem of solutions for an infinite system of ordinary differential equations. Besides, the existence and uniqueness of a D-pullback attractor {A(τ)}τ∈R is proved under general continuity assumptions and growth conditions but without Lipschitz hypothesis. The main idea is to show the pullback asymptotically upper semicompactness by deriving tails estimates of solutions and continuous dependence of solutions with respect to the initial data. There has not been published any work on the long-time behavior of tempered fractional nonlocal lattice systems with mixed delays, and this paper is devoted to fill this gap.

In this paper, we consider the asymptotic behavior of solutions of an evolution equation containing the non-autonomous p-Laplacian equation with polynomial growth nonlinearity of arbitrary order and dynamic boundary conditions driven by nonlinear colored noise. We first prove the existence of weak solutions by the Faedo–Galerkin method, but the uniqueness of solutions cannot be guaranteed due to the lack of Lipschitz continuity of diffusion and nonlinear terms. Then we establish the asymptotic compactness of the corresponding cocycle by Sobolev compactness embedding theorem and the measurability of the random attractor by proving the weak upper semi-continuity of the multi-valued non-autonomous cocycle. Finally, we prove the existence and uniqueness of pullback random attractor for the multi-valued non-autonomous cocycle generated by the solution operator.

Our main objective in this paper is to develop a second-order stochastic numerical method which generalizes the well-known deterministic TR-BDF2 scheme. Since most stochastic techniques used for approximating the solution of a stochastic differential equation may have lower order compared to the deterministic case, we have elaborated a scheme which not only preserves the second-order accuracy of the original scheme in the stochastic framework, but also its $A$-stability. Once we obtain the scheme and prove its second-order accuracy and $A$-stability, which is not a trivial task, we also state a result concerning its $MS$-stability. This concept is also analyzed for different parameter ranges in our scheme and the It{ô}--Taylor approximation of order 2, revealing scenarios where, for certain time step sizes, the developed method is $MS$-stable while the It{ô}--Taylor one is not. This concept is really useful to tackle slow-fast problems such as stiff ones, which we aim to explore further in future work. Finally, we validate the theoretical results with some academic test cases.