{"title":"The nonclassical diffusion equations with time-dependent memory kernels and a new class of nonlinearities","authors":"L. T. Thuy, N. Toan","doi":"10.1017/S0017089522000027","DOIUrl":null,"url":null,"abstract":"Abstract In this study, we consider the nonclassical diffusion equations with time-dependent memory kernels \n\\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*}\n on a bounded domain \n$\\Omega \\subset \\mathbb{R}^N,\\, N\\geq 3$\n . Firstly, we study the existence and uniqueness of weak solutions and then, we investigate the existence of the time-dependent global attractors \n$\\mathcal{A}=\\{A_t\\}_{t\\in\\mathbb{R}}$\n in \n$H_0^1(\\Omega)\\times L^2_{\\mu_t}(\\mathbb{R}^+,H_0^1(\\Omega))$\n . Finally, we prove that the asymptotic dynamics of our problem, when \n$k_t$\n approaches a multiple \n$m\\delta_0$\n of the Dirac mass at zero as \n$t\\to \\infty$\n , is close to the one of its formal limit \n\\begin{equation*}u_{t} - \\Delta u_{t} - (1+m)\\Delta u + f( u) = g. \\end{equation*}\n The main novelty of our results is that no restriction on the upper growth of the nonlinearity is imposed and the memory kernel \n$k_t(\\!\\cdot\\!)$\n depends on time, which allows for instance to describe the dynamics of aging materials.","PeriodicalId":50417,"journal":{"name":"Glasgow Mathematical Journal","volume":"64 1","pages":"716 - 733"},"PeriodicalIF":0.5000,"publicationDate":"2022-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Glasgow Mathematical Journal","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1017/S0017089522000027","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 2
Abstract
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.
期刊介绍:
Glasgow Mathematical Journal publishes original research papers in any branch of pure and applied mathematics. An international journal, its policy is to feature a wide variety of research areas, which in recent issues have included ring theory, group theory, functional analysis, combinatorics, differential equations, differential geometry, number theory, algebraic topology, and the application of such methods in applied mathematics.
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