Polynomial Growth Of Arbitrary Order Research Articles

Asymptotic behavior of solutions to nonclassical diffusion equations with degenerate memory and a time-dependent perturbed parameter

This article concerns the asymptotic behavior of solutions for a class of nonclassical diffusion equation with time-dependent perturbation coefficient and degenerate memory. We prove the existence and uniqueness of time-dependent global attractors in the family of time-dependent product spaces, by applying the operator decomposition technique and the contractive function method. Then we study the asymptotic structure of time-dependent global attractors as \(t\to \infty\). It is worth noting that the memory kernel function satisfies general assumption, and the nonlinearity \(f\) satisfies a polynomial growth of arbitrary order. For more information see https://ejde.math.txstate.edu/Volumes/2024/22/abstr.html

Long-time behavior of nonclassical diffusion equations with memory on time-dependent spaces

This paper aims to study the long-time behavior of nonclassical diffusion equation with memory and disturbance parameters on time-dependent space. By using the contractive process method on the family of time-dependent spaces and operator decomposition technique, the existence of pullback attractors is first proved. Then the upper semi-continuity of pullback attractors with respect to perturbation parameter ν in M t is obtained. It’s remarkable that the nonlinearity f satisfies the polynomial growth of arbitrary order.

Mean attractors and invariant measures of locally monotone and generally coercive SPDEs driven by superlinear noise

We study the global solvability, mean attractors and invariant measures for an abstract locally monotone and generally coercive SPDEs driven by infinite-dimensional superlinear noise defined in a dual space of intersection of finitely many Banach spaces. The main feature of this abstract system is that it covers a larger class of fundamental models which are included or not included previously. Under an extended locally monotone variational setting, we establish the global well-posedness, Itô's energy equality and existence of mean random attractors in some high-order Bochner spaces. The existence, uniqueness, support, (high-order and exponential) moment estimates, ergodicity, (pointwise and Wasserstein-type) exponentially mixing and asymptotic stability of invariant measures and evolution systems of measures are discussed for autonomous and nonautonomous stochastic equations. A stopping time technique is used to prove the convergence of solutions in probability in order to overcome the difficulty caused by the local monotonicity and superlinear growth of the coefficients. Our abstract results and unified methods are expected to be applied to various types of SPDEs like 2D Navier-Stokes equations, 2D MHD equations, 2D magnetic Bénard problem, Burgers type equations, 3D Leray α-model, convective Brinkman-Forchheimer equations, fractional (s,p)-Laplacian equations with monotone nonlinearities of polynomial growth of arbitrary order, and others.

Uniform attractors for nonclassical diffusion equations with perturbed parameter and memory

In this paper, we consider the long-time behavior of the nonclassical diffusion equation with perturbed parameter and memory on a bounded domain Ω⊂Rn(n≥3). The main feature of this model is that the equation contains a dissipative term with perturbation parameters −νΔu and the nonlinearity f satisfies the polynomial growth of arbitrary order. By using the nonclassical operator method and a new analytical method (or technique) (Lemma 2.7), the existence and regularity of uniform attractors generated for this equation are proved. Furthermore, we also get the upper semicontinuity of the uniform attractors when the perturbed parameter ν → 0. It is remarkable that if ν = 0, we can get the same conclusion as in the works of Toan et al. [Acta Appl. Math. 170, 789–822 (2020)] and Conti et al. [Commun. Pure Appl. Anal. 19, 2035–2050 (2020)], but the nonlinearity is critical.

Pullback attractors for p-Laplacian equations with delays

In this paper, we consider the p-Laplacian equations with hereditary effects and the nonlinear term f satisfying the polynomial growth of arbitrary order q − 1 (q ≥ 2). We analyze the well-posedness of solutions and establish the existence of the (CL2(Ω),CL2(Ω)) and (CL2(Ω),CLq(Ω))-pullback attractors by applying the idea of the bi-spaces and the asymptotic a priori estimate method.

Upper semicontinuity of attractors for nonclassical diffusion equations with arbitrary polynomial growth

In this paper, we mainly investigate upper semicontinuity and regularity of attractors for nonclassical diffusion equations with perturbed parameters ν and the nonlinear term f satisfying the polynomial growth of arbitrary order p-1 (p geq 2). We extend the asymptotic a priori estimate method (see (Wang et al. in Appl. Math. Comput. 240:51–61, 2014)) to verify asymptotic compactness and upper semicontinuity of a family of semigroups for autonomous dynamical systems (see Theorems 2.2 and 2.3). By using the new operator decomposition method, we construct asymptotic contractive function and obtain the upper semicontinuity for our problem, which generalizes the results obtained in (Wang et al. in Appl. Math. Comput. 240:51–61, 2014). In particular, the regularity of global attractors is obtained, which extends and improves some results in (Xie et al. in J. Funct. Spaces 2016:5340489, 2016; Xie et al. in Nonlinear Anal. 31:23–37, 2016).

Exponential attractors for nonclassical diffusion equations with arbitrary polynomial growth nonlinearity

<abstract><p>In this paper, the dynamical behavior of the nonclassical diffusion equation is investigated. First, using the asymptotic regularity of the solution, we prove that the semigroup $ \{S(t)\}_{t\geq 0} $ corresponding to this equation satisfies the global exponentially $ \kappa- $dissipative. And then we estimate the upper bound of fractal dimension for the global attractors $ \mathscr{A} $ for this equation and $ \mathscr{A}\subset H^1_0(\Omega)\cap H^2(\Omega) $. Finally, we confirm the existence of exponential attractors $ \mathscr{M} $ by validated differentiability of the semigroup $ \{S(t)\}_{t\geq 0} $. It is worth mentioning that the nonlinearity $ f $ satisfies the polynomial growth of arbitrary order.</p></abstract>

Long-Time Dynamics of Stochastic Lattice Plate Equations with Nonlinear Noise and Damping

In this article we investigate the global existence as well as long-term dynamics for a wide class of lattice plate equations on the entire integer set with nonlinear damping driven by infinite-dimensional nonlinear noise. The well-posedness of the system is established for a class of nonlinear drift functions of polynomial growth of arbitrary order as well as locally Lipschitz continuous diffusion functions depending on time. Both existence and uniqueness of weak pullback mean random attractors are established for the non-autonomous system when the growth rate of the drift function is almost linear. In addition, the existence of invariant measures for the autonomous system is also established in $$\ell ^2\times \ell ^2$$ when the growth rate of the drift function is superlinear. The main difficulty of deriving the tightness of a family of distribution laws of the solutions is surmounted in light of the idea of uniform tail-estimates on the solutions developed by Wang (Phys D 128:41–52, 1999).

Random dynamics of lattice wave equations driven by infinite-dimensional nonlinear noise

This paper is concerned with the global existence and random dynamics of non-autonomous stochastic second-order lattice systems driven by infinite-dimensional nonlinear noise defined on higher-dimensional integer sets. We first show the existence and uniqueness of mean square solutions to the equations when the nonlinear drift term has a polynomial growth of arbitrary order and the diffusion term is locally Lipschitz continuous. We then prove that the mean random dynamical system associated with the solution operator possesses a unique tempered weak pullback mean random attractor in a Bochner space under certain conditions. We finally establish the existence of invariant measures for the stochastic systems in \begin{document}$ \ell^2\times\ell^2 $\end{document} by showing the tightness of a family of distribution laws of solutions via the idea of uniform tail-estimates on the solutions.

Asymptotic behavior of fractional nonclassical diffusion equations driven by nonlinear colored noise on

This paper is concerned with the asymptotic behavior of the solutions of the fractional nonclassical diffusion equations driven by nonlinear colored noise defined on the entire space . We first establish the existence of energy equations for the solutions in with , and then prove the existence and uniqueness of pullback random attractors in when the nonlinear drift and diffusion terms have polynomial growth of arbitrary order. In addition, for linear additive noise, we show the upper semi-continuity of these attractors as the correlation time of the colored noise approaches zero. The idea of energy equations due to Ball is employed to establish the pullback asymptotic compactness of the solutions in in order to overcome the weak dissipativeness of the equation as well as the non-compactness of Sobolev embeddings on unbounded domains. The result of this paper is new even in the space when the fractional Laplace operator reduces to the standard Laplace operator.

$L^{p}$-pullback attractors for non-autonomous reaction-diffusion equations with delays

In this paper, we consider the non-autonomous reaction-diffusion equations with hereditary effects and the nonlinear term $f$ satisfying the polynomial growth of arbitrary order $p-1$ $(p\geq2)$. The delay term may be driven by a function with very weak assumptions, namely, just measurability. We extend the asymptotic \emph{a priori} estimate method (see \cite{ZYS06}) to our problem and establish a new existence theorem for the pullback attractors in $C_{L^{p}(\Omega)}$ $(p> 2)$ (see Theorem \ref{t2.5}), which generalizes the results obtained in \cite{GM14}.

Asymptotic behavior of non-autonomous fractional stochastic [formula omitted]-Laplacian equations

This article is concerned with the asymptotic behavior of solutions of a wide class of non-autonomous, non-local, fractional, stochastic p-Laplacian equations driven by multiplicative white noise. The time-dependent nonlinear drift term of the equation has a polynomial growth of arbitrary order in its third component which is allowed to be greater than the exponent p. We first employ the Faedo–Galerkin method to prove the well-posedness of the equation in an appropriate Hilbert space. We then establish the existence, uniqueness and periodicity of tempered pullback random attractors for the equations. The upper semi-continuity of these attractors is also derived as the density of noise tends to zero. The results of this paper are new even when the stochastic equation reduces to the deterministic one.

Pullback attractors for the non-autonomous reaction-diffusion equations in RN

In this paper, we prove the existence of the pullback attractors in L2(RN), Lp(RN), and H1(RN) for a nonlinear reaction-diffusion equation in unbounded domains, in which the nonlinearity f satisfies the polynomial growth of arbitrary order p − 1(p > 2) and the external forcing g∈Lloc2(R;L2(RN)) only subjects to the weaker integrability condition ∫−∞teαs‖g(s)‖22ds.

Dynamics for a stochastic degenerate parabolic equation

In this paper we consider the long-time behavior of a class of stochastic degenerate parabolic equations involving an operator which is X-elliptic with respect to a family of locally Lipschitz continuous vector fields X={X1,X2,…,Xm˜}. The nonlinearity satisfies a dissipative condition with polynomial growth of arbitrary order p≥2. The existence of the random attractor in L2(O), higher-order integrability and continuity in H are established.

Continuity and pullback attractors for a non-autonomous reaction–diffusion equation in [formula omitted

In this paper, we study the dynamics of a non-autonomous reaction–diffusion equation in RN with the nonlinearity f satisfying the polynomial growth of arbitrary order p−1(p≥2). Firstly, we prove the existence of the pullback Dα-attractor in L2(RN). Secondly, we use a scheme to establish some new estimates, higher-order integrability of the difference of the solutions near the initial time, instead of using the usual estimates about higher regularities and higher-order integrability of solutions. Thirdly, we verify that the usual (L2(RN),L2(RN))-pullback Dα-attractor indeed can pullback attract the Dα-class in L2+δ-norm for any δ∈[0,∞) and H1-norm, and the solutions of the equation in H1(RN) are continuous with respect to initial data. Finally, we obtain the pullback Dα-attractor in Lp(RN) and H1(RN) as an application of the higher-order integrability and the continuity respectively.

Uniform Attractors for Nonclassical Diffusion Equations with Memory

We introduce a new method (or technique), asymptotic contractive method, to verify uniform asymptotic compactness of a family of processes. After that, the existence and the structure of a compact uniform attractor for the nonautonomous nonclassical diffusion equation with fading memory are proved under the following conditions: the nonlinearityfsatisfies the polynomial growth of arbitrary order and the time-dependent forcing termgis only translation-bounded inLloc2(R;L2(Ω)).

Pullback attractors for nonclassical diffusion equations with delays

In this paper, we prove the existence of pullback attractors in CH01(Ω) for a nonclassical diffusion equation with delay term g(t, ut) which contains some hereditary characteristics. We consider two types of nonlinearity f: one is the case of critical growth and the other one is the polynomial growth of arbitrary order p − 1(p ≥ 2).

Dynamics for a stochastic reaction–diffusion equation with additive noise

In this paper, we present a new scheme to study the dynamics of a stochastic reaction–diffusion equation with the nonlinearity satisfying a dissipative condition with polynomial growth of arbitrary order p⩾2. Firstly we use this scheme to establish some new estimates, higher-order integrability of the difference of the solutions near the initial time, instead of using the usual estimates about higher regularities and higher-order integrability of solutions. Secondly we verify that the attraction in the usual (L2(Q),L2(Q)) D-pullback random attractor indeed can be L2+δ-norm for any δ∈[0,∞); and the solutions of the equation are continuous in H01(Q) with respect to initial data. Thirdly we obtain the existence of pullback random attractor in H01(Q) as an application of the continuity. Even for the corresponding deterministic cases, the results and methods introduced in this paper are new.

Long-term analysis of degenerate parabolic equations in ℝ N

Longtime behavior of degenerate equations with the nonlinearity of polynomial growth of arbitrary order on the whole space ℝ N is considered. By using l-trajectories methods, we proved that weak solutions generated by degenerate equations possess an (L 2 (ℝ N ), L loc 2 (ℝ N ))-global attractor. Moreover, the upper bounds of the Kolmogorov ɛ-entropy for such global attractor are also obtained.

L p -uniform attractor for nonautonomous reaction-diffusion equations in unbounded domains

The goal of this paper is to consider the asymptotic behavior of solutions of nonautonomous classical reaction-diffusion equations in unbounded domains with nonlinearity having a polynomial growth of arbitrary order. The existence and structure of a uniform attractor are obtained in the spaces L2(Rn) and Lp(Rn), respectively.