Continuity and pullback attractors for a non-autonomous reaction-diffusion equation in

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• 作者：Kaixuan Zhu^a ; ^{zhukx12@163.com" class="auth_mail" title="E-mail the corresponding author} ; Feng Zhou^a ; ^b ; ^{zhouf13@lzu.edu.cn" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Reaction&ndash ; diffusion equation ; Higher-order integrability ; Continuity ; Pullback attractors
• 刊名：Computers & Mathematics with Applications
• 出版年：2016
• 出版时间：May 2016
• 年：2016
• 卷：71
• 期：10
• 页码：2089-2105
• 全文大小：454 K

文摘

In this paper, we study the dynamics of a non-autonomous reaction–diffusion equation in R^N${\mathbb{R}}^N$ with the nonlinearity f$f$ satisfying the polynomial growth of arbitrary order p−1(p≥2)$p-1(p\ge 2)$. Firstly, we prove the existence of the pullback D_α${\mathcal{D}}_{\alpha }$-attractor in L²(R^N)$L^2\left({\mathbb{R}}^N\right)$. 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(R^N),L²(R^N))$(L^2\left({\mathbb{R}}^N\right),L^2\left({\mathbb{R}}^N\right))$-pullback D_α${\mathcal{D}}_{\alpha }$-attractor indeed can pullback attract the D_α${\mathcal{D}}_{\alpha }$-class in L^2+δ$L^{2+\delta }$-norm for any δ∈[0,∞)$\delta \in [0,\infty )$ and H¹$H^1$-norm, and the solutions of the equation in H¹(R^N)$H^1\left({\mathbb{R}}^N\right)$ are continuous with respect to initial data. Finally, we obtain the pullback D_α${\mathcal{D}}_{\alpha }$-attractor in L^p(R^N)$L^p\left({\mathbb{R}}^N\right)$ and H¹(R^N)$H^1\left({\mathbb{R}}^N\right)$ as an application of the higher-order integrability and the continuity respectively.