In this paper, we consider the strongly damped wave equations with hereditary effects and the nonlinearity f satisfying critical growth. The delay term g(t, ut) may be driven by a function with very weak assumptions, namely measurability. We prove the existence of pullback D-attractors in CH & oacute; (ohm) & times; CL2(ohm) by applying the decomposition technique.

This work studies the long-time dynamics of a Lamé system with supercritical and super-supercritical nonlinearities g and f. A key analytical challenge arises from the interplay between the gradient-divergence term (μ + λ)∇ div u and the nonlinearities, especially due to the super-supercritical growth of both f and g. To address this, we combine evolutionary systems theory with energy methods and the quasi-stability approach. Our main results establish the existence and structure of a compact global attractor, and in the supercritical case, we further construct an exponential attractor. These results yield a comprehensive characterization of the asymptotic behavior of the Lamé system.

We study the dynamics of a wave model with nonlinear damping J(||u||2)g partial derivative tu and a nonlinear term f(u), assuming that the growth rates of both f and g are governed by a quintic critical exponent. In this framework, we establish the existence and structural properties of weak and strong attractors for the solution semigroup associated with this equation. Our results extend and refine previous work in the literature, which focuses only on subquintic exponents.

This article presents a new scheme for studying the dynamics of a quintic wave equation with nonlocal weak damping in a 3D smooth bounded domain. As an application, the existence and structure of weak, strong, and exponential attractors for the solution semigroup of this equation are obtained. The investigation sheds light on the well-posedness and long-term behavior of nonlinear dissipative evolution equations with nonlinear damping and critical nonlinearity.

We establish the existence of a time-dependent exponential smooth attractor for a weakly damped wave equations with sup-cubic nonlinearity and hereditary effects in a 3D bounded domain. Our methods are mainly based on the recent extension of Strichartz estimates to the case of bounded domains, a more general criterion constructed by the quasi-stable technique and the standard bootstrapping technique.

With the widespread application of spatial data in fields like econometrics and geographic information science, the methods to enhance the robustness of spatial econometric model estimation and variable selection have become a central focus of research. In the context of the spatial error model (SEM), this paper introduces a variable selection method based on exponential square loss and the adaptive lasso penalty. Due to the non-convex and non-differentiable nature of this proposed method, convex programming is not applicable for its solution. We develop a block coordinate descent algorithm, decompose the exponential square component into the difference of two convex functions, and utilize the CCCP algorithm in combination with parabolic interpolation for optimizing problem-solving. Numerical simulations demonstrate that neglecting the spatial effects of error terms can lead to reduced accuracy in selecting zero coefficients in SEM. The proposed method demonstrates robustness even when noise is present in the observed values and when the spatial weights matrix is inaccurate. Finally, we apply the model to the Boston housing dataset.

We study the long-time dynamics of a wave equation with nonlocal weak damping, nonlocal weak anti-damping and sup-cubic nonlinearity. Based on the Strichartz estimates in a bounded domain, we obtain the global well-posedness of the Shatah–Struwe solutions. To overcome the difficulties brought by the nonlinear weak damping term, we present a new-type Gronwall’s lemma to obtain the dissipative for the Shatah–Struwe solutions semigroup of this equation. Finally, we establish the existence of a time-dependent exponential attractor with the help of a more general criteria constructed by the quasi-stable technique.

We consider dynamics of a semilinear heat equation on time-varying domains with lower regular forcing term. Instead of requiring the forcing term f(· ) to satisfy ∫ _-∞^te^λ sf(s)^2_L^2 ds<∞ for all t∈ℝ , we show that the solutions of a semilinear heat equation on time-varying domains are continuous with respect to initial data in H^1 topology and the usual (L^2,L^2) pullback 𝒟_λ -attractor indeed can attract in the H^1 -norm, provided that ∫ _-∞^te^λ sf(s)^2_H^-1(𝒪_s) ds< ∞ and f∈ L^2_loc(ℝ,L^2(𝒪_s)) .

We present a comprehensive investigation of the long-term dynamics generated by a semilinear wave equation with time-dependent coefficients and quintic nonlinearity on a bounded domain subject to Dirichlet boundary conditions. By employing rescaling techniques for time and utilizing the Strichartz estimates applicable to bounded domains, we initially study the global well-posedness of the Shatah–Struwe (S–S) solutions. Subsequently, we establish the existence of a uniform weak global attractor consisting of points on complete bounded trajectories through an approach based on evolutionary systems. Finally, we prove that this uniformly weak attractor is indeed strong by means of a backward asymptotic a priori estimate and the so-called energy method. Moreover, the smoothness of the obtained attractor is also shown with the help of a decomposition technique.

We present a comprehensive investigation of the long-term dynamics generated by a semilinear wave equation with time-dependent coefficients and quintic nonlinearity on a bounded domain subject to Dirichlet boundary conditions. By employing rescaling techniques for time and utilizing the Strichartz estimates applicable to bounded domains, we initially study the global well-posedness of the Shatah–Struwe (S–S) solutions. Subsequently, we establish the existence of a uniform weak global attractor consisting of points on complete bounded trajectories through an approach based on evolutionary systems. Finally, we prove that this uniformly weak attractor is indeed strong by means of a backward asymptotic a priori estimate and the so-called energy method. Moreover, the smoothness of the obtained attractor is also shown with the help of a decomposition technique.

This paper is concerned with a predator–prey model with prey-taxis and linear prey harvesting under the homogeneous Neumann boundary condition. The stability of the unique positive constant solution of the predator–prey model without prey-taxis is derived. Also, the emergence of Hopf bifurcation is concluded by choosing the proper Hopf bifurcation parameters. By the center manifold theorem and normal form, we compute the direction of Hopf bifurcation and the stability of the bifurcating solution. Moreover, the stationary pattern with prey-taxis is investigated. The conclusions show that prey harvesting and prey-taxis can enrich the dynamics.

In this paper, we consider a non-autonomous reaction-diffusion equation with hereditary effects and the nonlinearity f satisfying the polynomial growth of arbitrary p − 1 ( p ⩾ 2) order. We employ the asymptotic a priori estimate method (see (J. Differential Equations 223 (2006) 367–399)) to our problem and establish an existence criterion for the ( C L 2 ( Ω ) , C L p ( Ω ) )-uniform (w.r.t σ ∈ Σ) attractors (see Theorem 2.18). Then, we obtain the ( C L 2 ( Ω ) , C L p ( Ω ) ) and ( C L 2 ( Ω ) , C H 0 1 ( Ω ) )-uniform (w.r.t σ ∈ Σ) attractors by applying the existence criterion and the uniform (w.r.t σ ∈ Σ) Condition (C) respectively.

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.

该文考虑带有时滞项的复Ginzburg-Landau方程解的适定性和拉回吸引子的存在性,其中非线性项满足任意p-1(p＞2)次多项式增长.利用收缩函数方法验证解过程{U(t,τ)}t≥τ的紧性,得到CL2(Ω)中拉回吸引子的存在性.

In this paper,the long-time behavior of the following complex Ginzburg-Landau equation with p-Laplacian?Tu-(λ+iα)Δpu+(κ+iβ)|u|q-2u-γu=f(t)on time-varying domains without any upper re-striction on q≥2 under the assumptions (α/λ,β/κ)∈S1 (1/Cp)∩S (1/Cq ,1/Cq) has been studied.The existence and uniqueness of a variational solution satisfying energy equality,under the assumption that the spatial do-mains are bounded and increase with time has been proved.Moreover,the D-pullback attractor for thenon-au-tonomous dynamical system generated by this class of solutions has been established.

In this paper, based on the notation of time-dependent attractors introduced by Conti, Pata and Temam in (J. Differ. Equ. 255:1254–1277, 2013), we prove the existence of time-dependent global attractors in $\mathcal{H}_{t}$ for a class of nonclassical reaction–diffusion equations with the forcing term $g(x)\in H^{-1}(\varOmega )$ and the nonlinearity f satisfying the polynomial growth of arbitrary $p-1$ ($p\geq 2$) order, which generalizes the results obtained in (Appl. Anal. 94:1439–1449, 2015) and (Bound. Value Probl. 2016: 10, 2016).

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.

In this paper, we consider the weakly damped wave equations with hereditary effects, and the nonlinearity f satisfies critical growth. The delay term g(t, ut) may be driven by a function with very weak assumptions, namely, just measurability. We analyze the well-posedness of solutions and verify the existence of the pullback D-attractor in C H 0 1 (Ω)×C L 2 (Ω)by constructing the energy functional and combining with the idea of the contractive function.

In this article, we continue the study of the dynamics of the following complex Ginzburg-Landau equation ∂tu − (λ + iα)Δu + (κ + iβ)|u|p−2u − γu = f(t) on non-cylindrical domains. We assume that the spatial domains are bounded and increase with time, which is different from the diffeomorphism case presented in Zhou and Sun [Discrete Contin. Dyn. Syst., Ser. B 21, 3767–3792 (2016)]. We develop a new penalty function to establish the existence and uniqueness of a variational solution satisfying energy equality as well as some energy inequalities and prove the existence of a D-pullback attractor for the non-autonomous dynamical system generated by this class of solutions.

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.