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Zeitschrift für Analysis und ihre Anwendungen


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Volume 37, Issue 3, 2018, pp. 299–314
DOI: 10.4171/ZAA/1615

Published online: 2018-07-04

Averaging of Nonclassical Diffusion Equations with Memory and Singularly Oscillating Forces

Cung The Anh[1], Dang Thi Phuong Thanh[2] and Nguyen Duong Toan[3]

(1) Hanoi National University of Education, Vietnam
(2) Hung Vuong University, Viet Tri, Viet Nam
(3) Haiphong University, Vietnam

We consider for $\rho \in [0,1)$ and $\varepsilon > 0$, the following nonclassical diffusion equations with memory and singularly oscillating external force $$u_t -\Delta u_t - \Delta u - \int_0^\infty \kappa (s) \Delta u(t-s)ds+ f(u) = g_0(t)+\varepsilon^{- \rho}g_1(t/\varepsilon),$$ together with the averaged equation $$u_t - \Delta u_t - \Delta u - \int_0^\infty \kappa (s) \Delta u(t-s)ds+ f( u) = g_{0}( t)$$ formally corresponding to the limiting case $\varepsilon=0$. Under suitable assumptions on the nonlinearity and on the external force, we prove the uniform (w.r.t. $\varepsilon$) boundedness as well as the convergence of the uniform attractor $\mathcal A^{\varepsilon}$ of the first equation to the uniform attractor $\mathcal A^{0}$ of the second equation as $\varepsilon \to 0^+$.

Keywords: Nonclassical diffusion equation, uniform attractor, memory, singularly oscillating force, boundedeness, convergence

Anh Cung The, Thanh Dang Thi Phuong, Toan Nguyen Duong: Averaging of Nonclassical Diffusion Equations with Memory and Singularly Oscillating Forces. Z. Anal. Anwend. 37 (2018), 299-314. doi: 10.4171/ZAA/1615