We first prove the existence and uniqueness of pullback randomattractors for abstract multi-valued non-autonomous randomdynamical systems. The standard assumption compactness thesesystems can be replaced by asymptoticcompactness. Then, we apply theory to handle a randomreaction-diffusion equation with memory or delay terms which canbe considered on complete past defined $\mathbb{R}^{-}$. Inparticular, do not assume solutions theseequations.

This paper is devoted to investigating the well-posedness and asymptotic behavior of a class stochastic nonlocal partial differential equations driven by nonlinear noise. First, existence weak martingale solution established using Faedo--Galerkin approximation an idea analogous Da Prato Zabczyk [Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992]. Second, we show uniqueness continuous dependence on initial values solutions above problem when there exist...

The relationship between random attractors and global for dynamical systems is studied. If a partial differential equation perturbed by an E-small term certain hypotheses are satisfied, the upper semicontinuity of obtalned as c goes to zero. results applied Navier-Stokes equations problem reaction-diffusion type, both additive white noise.

This paper aims to an present account of some problems considered in the past years Dynamical Systems, new research directions and also provide open problems.

In this work we present the existence and uniqueness of pullbackand random attractors for stochastic evolution equations withinfinite delays when solutions theseequations is not required. Our results are obtained by means ofthe theory set-valued dynamical systems theirconjugation properties.

This paper is concerned with the asymptotic behaviour of solutions to a class non-autonomous stochastic nonlinear wave equations dispersive and viscosity dissipative terms driven by operator-type noise defined on entire space $\mathbb {R}^n$ . The existence, uniqueness, time-semi-uniform compactness asymptotically autonomous robustness pullback random attractors are proved in $H^1(\mathbb {R}^n)\times H^1(\mathbb {R}^n)$ when growth rate nonlinearity has subcritical range, density suitably...

In this paper, we consider fractional nonclassical diffusion equations under two forms: initial value problem and terminal problem. For an problem, study local existence, uniqueness, continuous dependence of the mild solution. We also present a result on unique continuation blow-up alternative for solutions pseudo-parabolic equations. show well-posedness our in case <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0 greater-than...

We consider the terminal value problem (or called final problem, initial inverse backward in time problem) of determining value, a general class time-fractional wave equations with Caputo derivative, from given value. are concerned existence, regularity solutions upon Under several assumptions on nonlinearity, we address and show well-posedness (namely, uniqueness, continuous dependence) for problem. Some results mild solution its derivatives first fractional orders also derived. The...

This work delves into the intricate realm of epidemic modeling under influence unpredictable surroundings. By harnessing power white noise and Lévy noise, we construct a robust framework to capture behavioral characteristics COVID-19 amidst erratic changes in external environment. To enhance our comprehension dynamics coronavirus, conducted an investigation using stochastic SIQS model that incorporates dedicated compartment represent populations quarantine. Thanks techniques, account for...

Some results on the existence and uniqueness of solutions to Navier–;Stokes equations when external force contains some hereditary characteristics are proved.