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The nonclassical diffusion equations with time-dependent memory kernels and a new class of nonlinearities

Part of: Partial differential equations Diffusion and convection Representations of solutions

Published online by Cambridge University Press:  21 February 2022

Le Thi Thuy  and
Nguyen Duong Toan
Le Thi Thuy
Affiliation:
Department of Mathematics, Electric Power University, 235 Hoang Quoc Viet, Tu Liem, Hanoi, Vietnam e-mail: [email protected]
Nguyen Duong Toan
Affiliation:
Faculty of Mathematics and Natural Sciences, Haiphong University, 171 Phan Dang Luu, Kien An, Haiphong, Vietnam e-mail: [email protected]
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Abstract

In this study, we consider the nonclassical diffusion equations with time-dependent memory kernels

\begin{equation*} u_{t} - \Delta u_t - \Delta u - \int_0^\infty k^{\prime}_{t}(s) \Delta u(t-s) ds + f( u) = g \end{equation*}
on a bounded domain $\Omega \subset \mathbb{R}^N,\, N\geq 3$ . Firstly, we study the existence and uniqueness of weak solutions and then, we investigate the existence of the time-dependent global attractors $\mathcal{A}=\{A_t\}_{t\in\mathbb{R}}$ in $H_0^1(\Omega)\times L^2_{\mu_t}(\mathbb{R}^+,H_0^1(\Omega))$ . Finally, we prove that the asymptotic dynamics of our problem, when $k_t$ approaches a multiple $m\delta_0$ of the Dirac mass at zero as $t\to \infty$ , is close to the one of its formal limit
\begin{equation*}u_{t} - \Delta u_{t} - (1+m)\Delta u + f( u) = g. \end{equation*}
The main novelty of our results is that no restriction on the upper growth of the nonlinearity is imposed and the memory kernel $k_t(\!\cdot\!)$ depends on time, which allows for instance to describe the dynamics of aging materials.

MSC classification

Primary: 35B41: Attractors 35D30: Weak solutions 76R50: Diffusion
Secondary: 45K05: Integro-partial differential equations
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 64 , Issue 3 , September 2022 , pp. 716 - 733
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Glasgow Mathematical Journal Trust

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