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In this work, we focus on analyzing non-local and quasilinear variations of the Chafee-Infante problem, including one variant with delay. To tackle these challenges, we use a time reparameterization, enabling us to reframe them as semilinear problems. We present an abstract approach to semilinear equations, offering broad applicability and facilitating extension to diverse models. Our analysis encompasses several key aspects. Firstly, we address the existence (possibly non-unique) of mild solutions and their regularity, contingent upon an appropriate modulus of continuity on the nonlinearity. Additionally, we explore comparison results utilizing the variation of constants formula. Leveraging these comparisons, we establish global existence, under the supplementary condition that the nonlinearity adheres to a structural criterion and is complemented by uniform bounds provided that the nonlinearity exhibits dissipative behavior. Furthermore, we establish the existence of global/pullback attractor for the associated multivalued semigroup/process.

Esta tesis está dedicada a la exploración de atractores para sistema dinámicos. Nuestro análisis sigue dos enfoques distintos: uno más abstracto, centrándose en atractores no acotados, y otra investigación más aplicada que involucra el modelo Lotka-Volterra no autónomo. La sección dedicada al atractor no acotado se ocupa de establecer la existencia de un conjunto invariante y cerrado que atraiga conjuntos acotados en el caso de que existan ´orbitas que tienden al infinito en tiempo infinito, es decir, para semigrupos lentamente no disipativos. Siguiendo la línea de [CG92], definimos el concepto de atractor no acotado, establecemos condiciones para su existencia y demostramos que es el conjunto de ´orbitas acotadas en el pasado. Además, profundizamos en su estructura interna, descomponiéndolo en un conjunto de órbitas completamente acotadas y sus conexiones heteroclínicas con el infinito, y estudiamos el conjunto !-límite para este tipo de semigrupos. Continuando nuestro estudio de atractores no acotados, extendemos las técnicas al marco no autónomo, proporcionando condiciones para la existencia de un atractor pullback no acotado. Nuestro análisis abarca sus propiedades y estructura, incluido el conjunto pullback !-límite. Concluimos esta sección aplicando los resultados encontrados a la ecuación en derivadas parciales u1 = Au + f(u), demostrando la existencia del atractor no acotado bajo ciertas condiciones impuestas sobre A y f. Por otro lado, la sección dedicada al modelo Lotka-Volterra no autónomo se centra en establecer condiciones para la existencia y caracterización del atractor forward. Inicialmente consideramos que el sistema es completamente no autónomo, con el vector de tasa de crecimiento intrínseco, y la matriz de interacción entre especies como funciones dependientes del tiempo. Bajo ciertas condiciones impuestas a dicho vector, demostramos la existencia de una solución globalmente estable junto con sus conexiones heteroclínicas con las soluciones semie, This thesis is dedicated to the exploration of attractors for a dynamical system. Our analysis follows two distinct paths: a more abstract approach, focusing on unbounded attractors, and a more applied investigation involving the non-autonomous Lotka-Volterra model. The section devoted to the unbounded attractor is concerned with establishing the existence of an invariant and closed set that attracts bounded sets in case where there exists orbits that go to infinity in infinite time, that is, for slowly non-dissipative semigroups. Following the lines of [CG92], we define the concept of unbounded attractor, we give conditions for its existence and we prove that is the set of orbits bounded in the past. Additionally, we delve into its internal structure, decomposing it into a set of completely bounded orbits and their heteroclinic connections with infinity, and study the !-limit set for this type of semigroups. Continuing our exploration of unbounded attractors, we extend our methods to the nonautonomous framework, providing conditions for the existence of an unbounded pullback attractor. Our analysis encompasses its properties and structure, including the pullback !-limit set. We conclude this section by applying the derived results to the partial differential equation u1 = Au + f(u), demonstrating the existence of the unbounded attractor under certain conditions imposed on A and f. On the other hand, the section dedicated to the non-autonomous Lotka-Volterra model is focused on establishing conditions for the existence and characterization of the forward attractor. Initially, we consider the system to be entirely nonautonomous, with vector of intrinsic growth rate and matrix of interaction between species as time-dependent functions. Under certain conditions imposed on this vector, we demonstrate the existence of a globally stable solution along with its heteroclinic connections with the semistable solutions. Additionally, we establish that any solution not belongin

This thesis is concerned with the dynamics of deterministic systems with memory, as well as stochastic systems with nonlinear white (colored) noise, with or without delay. In the three parts of the thesis, we specifically study nonlocal semilinear degenerate heat equations in Chapter 1, random p-Laplace equations in Chapter 2-Chapter 3, and stochastic modified Swift-Hohenberg lattice systems in Chapter 4-Chapter 5. In Chapter 1, we consider the existence of global attractors of nonlocal semilinear degenerate heat equations with degenerate memory on a bounded domain. We use the Faedo-Galerkin method to prove the existence of solutions of the non-degenerate heat equation, and then use it to approximate the equation obtained by the Dafermos transform, which is subsequently combined with the properties of the defined associated operators to obtain the existence, uniqueness, and regularity of solutions to the original degenerate problem. In addition, we establish an autonomous dynamical system and thereafter show the existence of the global attractor of the original problem. In Chapter 2, we show the continuity of pullback random attractors for random p-Laplace equations driven by nonlinear colored noise on unbounded domains. We establish abstract results for the existence and residual dense continuity of a unique pullback random bi-spatial attractor. In its proof, we may consider the larger metric space of all closed bounded sets in regular space and will use the abstract Baire residual theorem and the Baire density theorem. After that, we justify the existence and the residual dense continuity of pullback random bi-spatial attractors of the random p-Laplace equation on both initial space and regular space. In Chapter 3, we prove the existence of pullback random attractors for random p-Laplace equations with nonlinear color noise and infinite delays on a bounded domain. To obtain the existence of weak solutions to the equation, we use the traditional Galerkin approximat

This paper is mainly concerned with a kind of fractional stochastic evolution equa-tions driven by L\'evy noise in a bounded domain. We first state the well-posedness of the problem viaiterative approximations and energy estimates. Then, the existence and uniqueness of weak pullbackmean random attractors for the equations are established by defining a mean random dynamical sys-tem. Next, we prove the existence of invariant measures when the problem is autonomous by meansof the fact thatH\gamma (\scrO ) is compactly embedded inL2(\scrO ) with\gamma \in (0,1). Moreover, the unique-ness of this invariant measure is presented, which ensures the ergodicity of the problem. Finally, alarge deviation principle result for solutions of stochastic PDEs perturbed by small L\'evy noise andBrownian motion is obtained by a variational formula for positive functionals of a Poisson randommeasure and Brownian motion. Additionally, the results are illustrated by the fractional stochasticChafee--Infante equations.

In this thesis we investigate the long time behavior of random dynamical systems associ- ated to several kinds of stochastic equations with delays in terms of stability for stationary solutions, weak pullback mean random attractors, random attractors and numerical attrac- tors. The thesis consists of three parts, where the rst part covers Chapters 1-3, the last two cover Chapters 4 and 5, respectively. Chapters 1-3 are devoted to the random dynamics of 3D Lagrangian-averaged Navier- Stokes equations with in nite delay in three cases. In Chapter 1 we consider the stability analysis of such systems in the case of bounded domains. We rst use Galerkin's approximations to prove the existence and uniqueness of solutions when the non-delayed external force is locally integrable and the delay terms are globally Lipschitz continuous with an additional assumption. We then prove the existence of a unique stationary solution to the corresponding deterministic equation via the Lax- Milgram and the Schauder theorems. The stability and asymptotic stability of stationary solutions (equilibrium solutions) are also established. The local stability of stationary so- lutions for general delay terms is carried out by using a direct method and then apply the abstract results to two kinds of in nite delays. It is worth mentioning that all conditions are general enough to include several kinds of delays, where we mainly consider unbounded variable delays and in nite distributed delays. As we know, it is still an open and challenging problem to obtain su cient conditions ensuring the exponential stability of solutions in case of unbounded variable delay. Fortunately, we obtained the exponential stability of stationary solutions in the case of in nite distributed delay. However, we are able to further investigate the asymptotic stability of stationary solutions in the case of unbounded variable delay by constructing suitable Lyapunov functionals. Besides, we proved the polynomial asymptotic, In this thesis we investigate the long time behavior of random dynamical systems associated to several kinds of stochastic equations with delays in terms of stability for stationary solutions, weak pullback mean random attractors, random attractors and numerical attractors. The thesis consists of three parts, where the first part covers Chapters 1-3, the last two cover Chapters 4 and 5, respectively. Chapters 1-3 are devoted to the random dynamics of 3D Lagrangian-averaged Navier-Stokes equations with infinite delay in three cases. In Chapter 1 we consider the stability analysis of such systems in the case of bounded domains. We first use Galerkin’s approximations to prove the existence and uniqueness of solutions when the non-delayed external force is locally integrable and the delay terms are globally Lipschitz continuous with an additional assumption. We then prove the existence of a unique stationary solution to the corresponding deterministic equation via the Lax-Milgram and the Schauder theorems. The stability and asymptotic stability of stationary solutions (equilibrium solutions) are also established. The local stability of stationary solutions for general delay terms is carried out by using a direct method and then apply the abstract results to two kinds of infinite delays. It is worth mentioning that all conditions are general enough to include several kinds of delays, where we mainly consider unbounded variable delays and infinite distributed delays. As we know, it is still an open and challenging problem to obtain su cient conditions ensuring the exponential stability of solutions in case of unbounded variable delay. Fortunately, we obtained the exponential stability of stationary solutions in the case of infinite distributed delay. However, we are able to further investigate the asymptotic stability of stationary solutions in the case of unbounded variable delay by constructing suitable Lyapunov functionals. Besides, we proved the polynomial asymptotic

The polynomial stability problem of stochastic delay differential equations has been studied in recent years. In contrast, there are relatively few works on stochastic partial differential equations with pantograph delay. The present paper is devoted to investigating large-time asymptotic properties of solutions for stochastic pantograph delay evolution equations with nonlinear multiplicative noise. We first show that the mild solutions of stochastic pantograph delay evolution equations with nonlinear multiplicative noise tend to zero with general decay rate (including both polynomial and logarithmic rates) in the pth moment and almost sure senses. The analysis is based on the Banach fixed point theorem and various estimates involving the gamma function. Moreover, by using a generalized version of the factorization formula and exploiting an approximation technique and a convergence analysis, we construct the nontrivial equilibrium solution, defined for t∈R∈ℝ, for stochastic pantograph delay evolution equations with nonlinear multiplicative noise. In particular, the uniqueness, Hölder regularity in time and general stability, in the pth moment and almost sure senses, of the nontrivial equilibrium solution are established.

In this work, the continuity with respect to the Hurst parameter of solutions to stochastic evolution equations is studied. Compared with recent studies on such continuity property, the model here is considered in a different point of view in which the equations are of SPDEs type, the solution and the fractional Brownian motion take value on a Hilbert space. The main contribution is to investigate the existence and stability of the solution with respect to the Hurst index in the space .

This paper is mainly concerned with the long-term random dynamics for the non-autonomous 3D globally modified Navier-Stokes equations with nonlinear colored noise. We first prove the existence of random attractors of the nonautonomous random dynamical system generated by the solution operators of such equations. Then we establish the existence of invariant measures supported on the random attractors of the underlying system. Random Liouville type theorem is also derived for such invariant measures. Moreover, we further investigate the limiting relationship of invariant measures between the above equations and the corresponding limiting equations when the noise intensity approaches to zero. In addition, we show the invariant measures of such equations with additive white noise can be approximated by those of the corresponding equations with additive colored noise as the correlation time of the colored noise goes to zero.

We present the global modification of the Ladyzhenskaya equations, for incompressible non-Newtonian fluids. This modification is through a cut-off function that multiplies the convective term of the equation and an additional artificial smoothing dissipation term as part of the viscous term of the equation. The goal of this work is the comparative analysis between the modified system and the non-modified system. Therefore, we show the existence and regularity of weak solutions, the existence of global attractors, the estimation of the fractal dimension of the global attractors, and finally, the relationship of the autonomous dynamics between the modified system and the non-modified system.

This paper aims to analyze the behavior of the solutions of a stochastic perturbed system with respect to the solutions of the stochastic unperturbed system. To prove our stability results, we have derived a new Gronwall–type inequality instead of the Lyapunov techniques, which makes it easy to apply in practice and it can be considered as a more general tool in some situations. On the one hand, we present sufficient conditions ensuring the global practical uniform exponential stability of SDEs based on Gronwall’s inequalities. On the other hand, we investigate the global practical uniform exponential stability with respect to a part of the variables of the stochastic perturbed system by using generalized Gronwall’s inequalities. It turns out that, the proposed approach gives a better result comparing with the use of a Lyapunov function. A numerical example is presented to illustrate the applicability of our results.

The aim of this paper is to prove the existence and smoothness of stable and unstable invariant manifolds for a stochastic delayed partial differential equation of parabolic type. The stochastic delayed partial differential equation is firstly transformed into a random delayed partial differential equation by a conjugation, which is then recast into a Hilbert space. For the auxiliary equation, the variation of constants formula holds and we show the existence of Lipschitz continuous stable and unstable manifolds by the Lyapunov-Perron method. Subsequently, we prove the smoothness of these invariant manifolds under appropriate spectral gap condition by carefully investigating the smoothness of auxiliary equation, after which, we obtain the invariant manifolds of the original equation by projection and inverse transformation. Eventually, we illustrate the obtained theoretical results by their application to a stochastic single-species population model.

This article studies the lattice long-wave-short-wave resonance equations in weighted spaces. The authors first prove the global well-posedness of the initial value problem and the existence of the pullback attractor for the process generated by the solution mappings in the weighted space. Then they establish that the process possesses a family of invariant Borel probability measures supported by the pullback attractor. Afterwards, they verify that this family of Borel probability measures satisfies the Liouville theorem and is a statistical solution of the lattice long-wave-shortwave resonance equations. Finally, they prove an upper bound of the Kolmogorov entropy of the statistical solution

This paper mainly considers the long-term behavior of p-Laplace equations with infinite delays driven by nonlinear colored noise. We firstly prove the existence of weak solutions to the equation, but the uniqueness of solutions cannot be guaranteed due to the lack of Lipschitz continuity conditions, and thus generate a multi-valued dynamical system. Moreover, the regularity of solutions is also proved. Then we prove the existence of a pullback attractor. Subsequently, the measurability of the pullback attractor and the multi-valued dynamical system are also proved.

This paper is mainly concerned with the asymptotic dynamics of nonautonomous stochastic 3D globally modified Navier-Stokes equations driven by nonlinear noise. Based on the well-posedness of such equations, we first show the existence and uniqueness of weak pullback mean random attractors. Then we investigate the existence of (periodic) invariant measures, the zero-noise limit of periodic invariant measures and their limit as the modification parameter N → N0 ∈ (0, +∞). Furthermore, under weaker conditions, we obtain the existence of invariant measures as well as their limiting behaviors when the external term is independent of time. Finally, by using weak convergence method, we establish the large deviation principle for the solution processes.

In this paper, the existence and the limiting behavior of periodic measures for the periodic stochastic modified Swift-Hohenberg lattice systems with variable delays are analyzed. We first prove the existence and uniqueness of global solution when the nonlinear T -periodic drift and diffusion terms are locally Lipchitz continuous and linearly growing. Then we show the existence of periodic measures of the system under some assumptions. Finally, by strengthening the assumptions, we prove that the set of all periodic measures is weakly compact, and we also show that every limit point of a sequence of periodic measures of the original system must be a periodic measure of the limiting system when the noise intensity tends to zero.

Existence and connection of numerical attractors for discrete-time p -Laplace lattice systems via the implicit Euler scheme are proved. The numerical attractors are shown to have an optimized bound, which leads to the continuous convergence of the numerical attractors when the graph of the nonlinearity closes to the vertical axis or when the external force vanishes. A new type of Taylor expansion without Fréchet derivatives is established and applied to show the discretization error of order two, which is crucial to prove that the numerical attractors converge upper semicontinuously to the global attractor of the original continuous-time system as the step size of the time goes to zero. It is also proved that the truncated numerical attractors for finitely dimensional systems converge upper semicontinuously to the numerical attractor and the lower semicontinuity holds in special cases.

In this paper we study the asymptotic behaviour of multivalued processes which are under the influence of impulsive action. We provide conditions to guarantee the existence of a pullback attractor and we illustrate the results with several examples.

In this paper we investigate stochastic dynamics and invariant measures for stochastic 3D Lagrangian-averaged NavierStokes (LANS) equations driven by infinite delay and additive noise. We first use the Galerkin approximations, a priori estimates and standard Gronwall lemma to show the well-posedness for the corresponding random equation, whose solution operators lead to the existence of a random dynamical system. Next, the asymptotic compactness for the random dynamical system is established via the Ascoli-Arzel`a theorem. Besides, we derive the existence of a global random attractor for the random dynamical system. Moreover, we prove that the random dynamical system is bounded and continuous with respect to the initial time. Eventually, we construct a family of invariant Borel probability measures, which is supported by the global random attractor.

The objective of this paper is to study a stochastic epidemiological model with infinite Lévy measure and relapse. Using stochastic tools, we prove the existence and uniqueness of global positive solution. Moreover, we also show the extinction and persistence in mean of the disease by the use of Kunita’s inequality instead of Burkholder–Davis–Gundy inequality for continuous diffusions. The numerical behavior of the considered model is analyzed to understand the impact of environmental transmission on the spread of human and zonotic tuberculosis in Morocco.

This paper is mainly concerned with limiting behaviors of invariant measures for neural field latticemodels in a random environment. First of all, we consider the convergence relation of invariantmeasures between the stochastic neural field lattice model and the corresponding deterministic modelin weighted spaces, and prove any limit of a sequence of invariant measures of such a lattice modelmust be an invariant measure of its limiting system as the noise intensity tends to zero. Then we aredevoted to studying the numerical approximation of invariant measures of such a stochastic neurallattice model. To this end, we first consider convergence of invariant measures between such a neurallattice model and the system with neurons only interacting with its n-neighborhood; then we furtherprove the convergence relation of invariant measures between the system with an n-neighborhood andits finite dimensional truncated system. By this procedure, the invariant measure of the stochasticneural lattice models can be approximated by the numerical invariant measure of a finite dimensionaltruncated system based on the backward Euler--Maruyama (BEM) scheme. Therefore, the invariantmeasure of a deterministic neural field lattice model can be observed by the invariant measure ofthe BEM scheme when the noise is not negligible.

The aim of this paper is to prove the existence and qualitative property of random attractors for a stochastic non-local delayed reaction–diffusion equation (SNDRDE) on a semi-infinite interval with a Dirichlet boundary condition at the finite end. This equation models the spatial–temporal evolution of the mature individuals for a two-stage species whose juvenile and adults both diffuse that lives on a semi-infinite domain and subject to random perturbations. By transforming the SNDRDE into a random evolution equation with delay, by means of a stationary conjugate transformation, we first establish the global existence and uniqueness of solutions to the equation, after which we show the solutions generate a random dynamical system. Then, we deduce uniform a priori estimates of the solutions and show the existence of bounded random absorbing sets. Subsequently, we prove the pullback asymptotic compactness of the random dynamical system generated by the SNDRDE with respect to the compact open topology, and hence obtain the existence of random attractors. At last, it is proved that the random attractor is an exponentially attracting stationary solution under appropriate conditions. The theoretical results are illustrated by application to the stochastic non-local delayed Nicholson’s blowfly equation.

In this paper, we investigate the classical chemostat model where the consumption function of the species, in both cases Monod and Haldane, is perturbed by real random fluctuations. Once the existence and uniqueness of non-negative global solution of the corresponding random systems is ensured, we prove the existence of a deterministic compact attracting set, whence we are able to find conditions to guarantee either the extinction or the persistence of the species, the most important aim in real applications. In addition, we depict several numerical simulations to illustrate the theoretical framework, standing out our contributions, providing the biological interpretation of every result and comparing with similar works in the literature.

A kind of nonlocal reaction-diffusion equations on an unbounded domain containing a fractional Laplacian operator is analyzed. To be precise, we prove the convergence of solutions of the equation governed by the fractional Laplacian to the solutions of the classical equation governed by the standard Laplacian, when the fractional parameter grows to 1. The existence of global attractors is investigated as well. The novelty of this paper is concerned with the convergence of solutions when the fractional parameter varies, which, as far as the authors are aware, seems to be the first result of this kind of problems in the literature.

The main objective of this paper is to obtain estimations of Hausdorff dimension as well as fractal dimension of global attractors and pullback attractors for both autonomous and nonautonomous functional differential equations (FDEs) in Banach spaces. New criterions for the finite Hausdorff dimension and fractal dimension of attractors in Banach spaces are firslty proposed by combining the squeezing property and the covering of finite subspace of Banach spaces, which generalize the method established in Hilbert spaces. In order to surmount the barrier caused by the lack of orthogonal projectors with finite rank, which is the key tool for proving the squeezing property of partial differential equations in Hilbert spaces, we adopt the state decomposition of phase space based on the exponential dichotomy of the studied FDEs to obtain similar squeezing property. The theoretical results are applied to a retarded nonlinear reaction-diffusion equation and a non-autonomous retarded functional differential equation in the natural phase space, for which explicit bounds of dimensions that do not depend on the entropy number but only depend on the spectrum of the linear parts and Lipschitz constants of the nonlinear parts are obtained.

In this paper, we establish theoretical results on the stability of random regular attractors. First, we introduce a backward regular attractor, which is a new type of attractor defined by a minimal backward pullback attracting set. We then establish an existence theorem for such an attractor, and prove it is long time stable. Eventually, we prove the long time stability of regular pullback random attractors. As an application, we consider stochastic non-autonomous Newton–Boussinesq equations with variable and distributed delays. Since solutions of the equations have no higher regularity, we prove their regular asymptotic compactness via the spectrum decomposition technique.

The aim of this paper is to propose a new method to construct pullback exponential attractors with explicit fractal dimensions for non-autonomous infinite-dimensional dynamical systems in Banach spaces. The approach is established by combining the squeezing properties and the covering of finite subspace of Banach spaces, which generalize the method established for autonomous systems in Hilbert spaces (Eden A, Foias C, Nicolaenko B, and Temam R Exponential attractors for dissipative evolution equations, Wiley, New York, 1994). The method is especially effective for non-autonomous partial functional differential equations for which phase space decomposition based on the exponential dichotomy of the linear part or variation techniques are available for proving squeezing property. The theoretical results are illustrated by applications to several specific non-autonomous partial functional differential equations, including a retarded reaction–diffusion equation, a retarded 2D Navier–Stokes equation and a retarded semilinear wave equation. The constructed exponential attractors possess explicit fractal dimensions which do not depend on the entropy number but only on some inner characteristics of the studied equations including the spectra of the linear part and the Lipschitz constants of the nonlinear terms and hence do not require the smooth embedding between two spaces in the previous work.

On the one hand, the primitive three-dimensional viscous equations for large-scale ocean and atmosphere dynamics are commonly used in weather and climate predictions. On the other hand, ever since the middle of the last century, it has been widely recognized that the climate variability exhibits long-time memory. In this paper, we first prove the global existence of weak solutions to the primitive equations of large-scale ocean and atmosphere dynamics with Caputo fractional time derivatives. Then we establish the existence of an absorbing set, which is positively invariant. Finally, an attractor (strictly speaking, the minimal attracting set containing all the limiting dynamics) is constructed for the time fractional primitive equations, which means that the present state of a system may have long-time influences on the states in far future. However, there was no work on the long-time behavior of the time fractional primitive equations and we fill this gap in this paper.

This paper is devoted to investigating mean dynamics and stability analysis for stochastic 3D Lagrangian-averaged Navier–Stokes (LANS) equations driven by infinite delay on unbounded domains. We first prove the existence of a unique solution to stochastic 3D LANS equations with infinite delay when the non-delayed external force is locally integrable, the delay term is globally Lipschitz continuous and the nonlinear diffusion term is locally Lipschitz continuous. This enables us to define a mean random dynamical system. Besides, we find that such a dynamical system possesses a unique weak pullback mean random attractor, which is a minimal, weakly compact and weakly pullback attracting set. Furthermore, we prove the existence and uniqueness of stationary solutions (equilibrium solutions) to the corresponding deterministic equation via the classical Galerkin method, the Lax–Milgram and the Brouwer fixed theorems. The stability properties of stationary solutions are also considered. By a direct approach, we first show the local stability of stationary solutions when the delay term has a general form and then apply the abstract results to two kinds of infinite delays. Second, we establish the exponential stability of stationary solutions in the case of unbounded distributed delay. Third, we investigate the asymptotic stability of stationary solutions in the case of unbounded variable delay by constructing appropriate Lyapunov functionals. Eventually, we discuss the polynomial asymptotic stability in the particular case of proportional delay.

In this article, we are interested in investigating the nonlocal nonlinear reaction– diffusion system with final conditions. This problem is called backward in time problem, or terminal value problem which is understood as redefining the previous distributions when the distribution data at the terminal observation are known. There are three main goals presented in this paper. First, we prove that the problem is ill-posed (often called as unstable property) in the sense of Hadamard. Our next propose is to provide a modified quasi-reversibility model to stabilize the ill-posed problem. Using some techniques and tools of Faedo–Galerkin method, we prove the existence of the unique weak solution of the regularized problem. Further, we investigate error estimates between the sought solution and the regularized solution in L2(Ω)− and H1(Ω)− norms. The final aim of this paper is to give some numerical results to demonstrate that our method is useful and effective.

In this article, we first prove some sufficient conditions guaranteeing the existence of invariant sample measures for random dynamical systems via the approach of global random attractors. Then we consider the two-dimensional incompressible Navier-Stokes equations with additive white noise as an example to show how to check the sufficient conditions for concrete stochastic partial differential equations. Our results generalize the Liouville type theorem to the random case and reveal that the invariance of the sample measures is a particular situation of the random Liouville type theorem

This paper is devoted to the asymptotic behavior of solutions to a non-autonomous stochastic wave equation with infinite delays. The nonlinear terms of the equation are not expected to be Lipschitz continuous, but only satisfy continuity assumptions along with growth conditions, under which the uniqueness of the solutions may not hold. Using the theory of multi-valued non-autonomous random dynamical systems, we prove the existence and measurability of a compact global pullback attractor.

In this paper, it is first addressed the well-posedness of weak solutions to a nonlocal partial differential equation with long time memory, which is carried out by exploiting the nowadays well-known technique used by Dafermos in the early 70’s. Thanks to this Dafermos transformation, the original problem with memory is transformed into a nondelay one for which the standard theory of autonomous dynamical system can be applied. Thus, some results about the existence of global attractors to the transformed problem are provided. Particularly, when the initial values have higher regularity, the solutions of both problems (the original and the transformed ones) are equivalent. Nevertheless, the equivalence of global attractors for both problems is still unsolved due to the lack of enough regularity of solutions in the transformed problem, it is therefore proved the existence of global attractors of the transformed problem. Eventually, it is highlighted how to proceed to obtain meaningful results about the original problem, without performing any transformation, but working directly with the original delay problem.

The Lyapunov approach is one of the most effective and efficient methods for the inves tigation of the stability of stochastic systems. Several authors analyzed the stability and stabilization of stochastic differential equations via Lyapunov techniques. Nevertheless, few results are concerned with the stability of stochastic systems based on the knowledge of the solution of the system explicitly. The originality of our work is to investigate the problem of stabilization of stochastic perturbed control–bilinear systems based on the explicit solution of the system by using the integral inequalities of the Gronwall type in particular Gami dov’s inequality. Namely, under some restrictions on the perturbed term, and based on the method of integral inequalities, we prove that the stochastic system can be stabilized by constant feedback. Further, we study the problem of stabilization of stochastic perturbed control affine systems based on the use of bilinear approximation. Different examples are provided to verify the effectiveness of the proposed results.