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Existence and upper semicontinuity of uniform attractors in $H^1(\mathbb {R}^N)$ for nonautonomous nonclassical diffusion equations

Volume 111 / 2014

Cung The Anh, Nguyen Duong Toan Annales Polonici Mathematici 111 (2014), 271-295 MSC: Primary 35B41; Secondary 35K70, 35D30. DOI: 10.4064/ap111-3-5

Abstract

We prove the existence of uniform attractors $\mathcal A_{\varepsilon }$ in the space $H^1(\mathbb {R}^N)$ for the nonautonomous nonclassical diffusion equation $$ u_t - \varepsilon \varDelta u_t - \varDelta u + f(x,u)+\lambda u = g(x,t),\hskip 1em \varepsilon \in [0,1]. $$ The upper semicontinuity of the uniform attractors $\{\mathcal A_{\varepsilon }\}_{\varepsilon \in [0,1]}$ at $\varepsilon = 0$ is also studied.

Authors

  • Cung The AnhDepartment of Mathematics
    Hanoi National University of Education
    136 Xuan Thuy, Cau Giay
    Hanoi, Vietnam
    e-mail
  • Nguyen Duong ToanDepartment of Mathematics
    Haiphong University
    171 Phan Dang Luu, Kien An
    Haiphong, Vietnam
    e-mail

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