摘要：
<jats:title>Abstract</jats:title><jats:p>In this paper, we mainly investigate upper semicontinuity and regularity of attractors for nonclassical diffusion equations with perturbed parameters<jats:italic>ν</jats:italic>and the nonlinear term<jats:italic>f</jats:italic>satisfying the polynomial growth of arbitrary order<jats:inline-formula><jats:alternatives><jats:tex-math>$p-1$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>p</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:math></jats:alternatives></jats:inline-formula>(<jats:inline-formula><jats:alternatives><jats:tex-math>$p \geq 2$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>p</mml:mi><mml:mo>≥</mml:mo><mml:mn>2</mml:mn></mml:math></jats:alternatives></jats:inline-formula>). 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).</jats:p>
期刊：
BOUNDARY VALUE PROBLEMS,2020年2020(1):1-14 ISSN：1687-2762
通讯作者：
Zhu, Kaixuan
作者机构：
[Zhu, Kaixuan] Hunan Univ Arts & Sci, Coll Math & Phys Sci, Hunan Prov Cooperat Innovat Ctr Construct & Dev D, Changde, Peoples R China.;[Xie, Yongqin] Changsha Univ Sci & Technol, Sch Math & Stat, Changsha, Peoples R China.;[Zhou, Feng] China Univ Petr East China, Coll Sci, Qingdao, Peoples R China.
通讯机构：
[Zhu, Kaixuan] H;Hunan Univ Arts & Sci, Coll Math & Phys Sci, Hunan Prov Cooperat Innovat Ctr Construct & Dev D, Changde, Peoples R China.
关键词：
Nonclassical reaction–diffusion equations;Polynomial growth of arbitrary order;Time-dependent global attractors
摘要：
<jats:title>Abstract</jats:title><jats:p>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 <jats:inline-formula><jats:alternatives><jats:tex-math>$\mathcal{H}_{t}$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>H</mml:mi><mml:mi>t</mml:mi></mml:msub></mml:math></jats:alternatives></jats:inline-formula> for a class of nonclassical reaction–diffusion equations with the forcing term <jats:inline-formula><jats:alternatives><jats:tex-math>$g(x)\in H^{-1}(\varOmega )$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>g</mml:mi><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo><mml:mo>∈</mml:mo><mml:msup><mml:mi>H</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo>(</mml:mo><mml:mi>Ω</mml:mi><mml:mo>)</mml:mo></mml:math></jats:alternatives></jats:inline-formula> and the nonlinearity <jats:italic>f</jats:italic> satisfying the polynomial growth of arbitrary <jats:inline-formula><jats:alternatives><jats:tex-math>$p-1$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>p</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:math></jats:alternatives></jats:inline-formula> (<jats:inline-formula><jats:alternatives><jats:tex-math>$p\geq 2$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>p</mml:mi><mml:mo>≥</mml:mo><mml:mn>2</mml:mn></mml:math></jats:alternatives></jats:inline-formula>) order, which generalizes the results obtained in (Appl. Anal. 94:1439–1449, 2015) and (Bound. Value Probl. 2016: 10, 2016).</jats:p>
摘要：
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 >= 2). The delay term may be driven by a function with very weak assumptions, namely, just measurability. We extend the asymptotic a priori estimate method (see [29]) to our problem and establish a new existence theorem for the pullback attractors in C-Lp (Omega) (p > 2) (see Theorem 2.12), which generalizes the results obtained in [12].
期刊：
Journal of Mathematical Physics,2019年60(3):032702 ISSN：0022-2488
通讯作者：
Zhu, Kaixuan
作者机构：
[Zhu, Kaixuan] Hunan Univ Arts & Sci, Coll Math & Phys Sci, Hunan Prov Cooperat Innovat Ctr Construct & Dev D, Changde 415000, Peoples R China.;[Xie, Yongqin] Changsha Univ Sci & Technol, Sch Math & Stat, Changsha 410114, Hunan, Peoples R China.;[Zhou, Feng] China Univ Petr East China, Coll Sci, Qingdao 266580, Shandong, Peoples R China.;[Li, Xin] Yanshan Univ, Sch Sci, Qinhuangdao 066004, Hebei, Peoples R China.
通讯机构：
[Zhu, Kaixuan] H;Hunan Univ Arts & Sci, Coll Math & Phys Sci, Hunan Prov Cooperat Innovat Ctr Construct & Dev D, Changde 415000, Peoples R China.
关键词：
integral equations;reaction-diffusion systems
摘要：
In this paper, we prove the existence of the pullback attractors in L-2 (R-N), L-p (R-N), and H-1 (R-N) 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 is an element of L-loc(2) (R; L-2 (R-N)) only subjects to the weaker integrability condition integral(t)(-infinity) e(alpha s)parallel to g(s)parallel to(2)(2)ds. Published under license by AIP Publishing.
作者机构：
[Zhu, Kai Xuan] Hunan Univ Arts & Sci, Hunan Prov Cooperat Innovat Ctr Construct & Dev D, Changde 415000, Peoples R China.;[Zhu, Kai Xuan] Hunan Univ Arts & Sci, Coll Math & Computat Sci, Changde 415000, Peoples R China.;[Xie, Yong Qin] Changsha Univ Sci & Technol, Sch Math & Stat, Changsha 410114, Hunan, Peoples R China.;[Zhou, Feng] China Univ Petr East China, Coll Sci, Qingdao 266580, Peoples R China.
通讯机构：
[Zhu, Kai Xuan] H;Hunan Univ Arts & Sci, Hunan Prov Cooperat Innovat Ctr Construct & Dev D, Changde 415000, Peoples R China.;Hunan Univ Arts & Sci, Coll Math & Computat Sci, Changde 415000, Peoples R China.
摘要：
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, u
t
) 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
$${C_{H_0^1\left( \Omega \right)}} \times {C_{{L^2}\left( \Omega \right)}}$$
by constructing the energy functional and combining with the idea of the contractive function.
摘要：
In this paper, long-time behavior of a class of non-autonomous viscoelastic equations with fading memory is investigated. We establish the existence of a compact uniform attractor together with its st
摘要：
Abstract: In this paper, we study the asymptotic behavior of solutions of third-order nonlinear functional differential equation with distributed delay. By using non-classical Riccati transformation, Young’s inequality and integral averaging, we establish some new sufficient conditions which ensure that every solution of this equation oscillated or converged to zero. Our results essentially improve and complement known results in the literature recently.#@#@#摘要: 本文研究一类具有分布时滞的三阶非线性泛函微分方程解的渐近行为，利用推广的Riccati变换和Young不等式，通过积分平均方法，获得了泛函微分方程一些新的振动性判据，改进和推广了最近文献中的一些结果。
摘要：
In this paper, a class of third-order quasi-linear differential equations with continuously distributed delay is studied. Applying the generalized Riccati transformation, integral averaging technique of Philos type and Young's inequality, a set of new criteria for oscillation or certain asymptotic behavior of nonoscillatory solutions of this equations is given. Our results essentially improve and complement some earlier publications.
期刊：
Journal of Mathematical Physics,2012年53(8):082703 ISSN：0022-2488
通讯作者：
Xie, Yongqin
作者机构：
[Xie, Yongqin; Zhu, Kaixuan] Changsha Univ Sci & Technol, Sch Math & Comp Sci, Changsha 410114, Hunan, Peoples R China.;[Sun, Chunyou; Zhu, Kaixuan] Lanzhou Univ, Sch Math & Stat, Lanzhou 730000, Peoples R China.
通讯机构：
[Xie, Yongqin] C;Changsha Univ Sci & Technol, Sch Math & Comp Sci, Changsha 410114, Hunan, Peoples R China.
关键词：
nonlinear dynamical systems;nonlinear equations;reaction-diffusion systems
摘要：
<jats:p>In this paper, we introduce a new class of functions satisfying spacial absolutely continuous (see Definition 3.1), denoted by \documentclass[12pt]{minimal}\begin{document}$L^{2}_{sac}(\mathbb {R};\mathbb {R}^{n})$\end{document}Lsac2(R;Rn), which are translation bounded but not normal (see [S. S. Lu, H. Q. Wu, and C. K. Zhong, “Attractors for non-autonomous 2D Navier-Stokes equations with normal external forces,” Discrete Contin. Dyn. Syst. A 13(3), 701−719 (2005)]10.3934/dcds.2005.13.701 and Definition 3.1) in \documentclass[12pt]{minimal}\begin{document}$L^{2}_{loc}(\mathbb {R};\mathbb {R}^{n})$\end{document}Lloc2(R;Rn). Then the asymptotic a priori estimate is applied to some nonlinear reaction-diffusion equations with external forces \documentclass[12pt]{minimal}\begin{document}$g(x,s)\in L^{2}_{sac}(\mathbb {R};\mathbb {R}^{n})$\end{document}g(x,s)∈Lsac2(R;Rn). We obtain the existence of uniform attractor together with its structure in the bi-spaces \documentclass[12pt]{minimal}\begin{document}$(L^{2}(\mathbb {R}^{n}), L^{2}(\mathbb {R}^{n}))$\end{document}(L2(Rn),L2(Rn)) and \documentclass[12pt]{minimal}\begin{document}$(L^{2}(\mathbb {R}^{n}), L^{p}(\mathbb {R}^{n}))(p&gt;2)$\end{document}(L2(Rn),Lp(Rn))(p&gt;2) without any restriction on the growing order of the nonlinear term.</jats:p>
期刊：
Journal of Mathematical Analysis and Applications,2007年336(1):54-69 ISSN：0022-247X
通讯作者：
Xie, Yongqin
作者机构：
[Xie, Yongqin] Lanzhou Univ, Sch Math & Stat, Lanzhou 730000, Peoples R China.;Changsha Univ Sci & Technol, Sch Math & Comp Sci, Changsha 410076, Hunan, Peoples R China.
通讯机构：
[Xie, Yongqin] L;Lanzhou Univ, Sch Math & Stat, Lanzhou 730000, Peoples R China.
关键词：
nonlinear evolution equation;global attractor;asymptotic a priori estimate
摘要：
In this paper, using a new method (or framework), we establish the existence of global attractors for a class nonlinear evolution equation in H-0(1)(Omega) x H-0(1)(Omega), where the nonlinear term f satisfies a critical exponential growth condition. (c) 2007 Elsevier Inc. All rights reserved.
