A Remark on Nonclassical Diffusion Equations with Memory

A Remark on Nonclassical Diffusion Equations with Memory The nonclassical diffusion equation with hereditary memory \begin{aligned} u_t-\Delta u_t-\Delta u-\int _0^\infty \kappa (s)\Delta u(t-s)\,\mathrm{d}s +\varphi (u)=f \end{aligned} u t - Δ u t - Δ u - ∫ 0 ∞ κ ( s ) Δ u ( t - s ) d s + φ ( u ) = f on a 3D bounded domain is considered, for a very general class of memory kernels $$\kappa$$ κ . Setting the problem both in the classical past history framework and in the more recent minimal state one, the related solution semigroups are shown to possess finite-dimensional regular exponential attractors. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematics and Optimization Springer Journals

A Remark on Nonclassical Diffusion Equations with Memory

, Volume 73 (1) – Feb 1, 2016
21 pages

/lp/springer_journal/a-remark-on-nonclassical-diffusion-equations-with-memory-2a1qCvIkFK
Publisher
Springer US
Subject
Mathematics; Calculus of Variations and Optimal Control; Optimization; Systems Theory, Control; Theoretical, Mathematical and Computational Physics; Mathematical Methods in Physics; Numerical and Computational Physics
ISSN
0095-4616
eISSN
1432-0606
D.O.I.
10.1007/s00245-015-9290-8
Publisher site
See Article on Publisher Site

Abstract

The nonclassical diffusion equation with hereditary memory \begin{aligned} u_t-\Delta u_t-\Delta u-\int _0^\infty \kappa (s)\Delta u(t-s)\,\mathrm{d}s +\varphi (u)=f \end{aligned} u t - Δ u t - Δ u - ∫ 0 ∞ κ ( s ) Δ u ( t - s ) d s + φ ( u ) = f on a 3D bounded domain is considered, for a very general class of memory kernels $$\kappa$$ κ . Setting the problem both in the classical past history framework and in the more recent minimal state one, the related solution semigroups are shown to possess finite-dimensional regular exponential attractors.

Journal

Applied Mathematics and OptimizationSpringer Journals

Published: Feb 1, 2016

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