Nonclassical diffusion with memory lacking instantaneous damping

Monica Conti; Filippo Dell'Oro; Vittorino Pata

doi:10.3934/cpaa.2020090

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2020, Volume 19, Issue 4: 2035-2050. Doi: 10.3934/cpaa.2020090

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Nonclassical diffusion with memory lacking instantaneous damping

Politecnico di Milano - Dipartimento di Matematica, Via Bonardi 9, 20133 Milano, Italy

^*Corresponding author
^*Corresponding author

Dedicated to our colleague and friend Professor Tomás Caraballo on the occasion of his sixtieth birthday

Received: February 28, 2019

Revised: April 30, 2019

Published: January 2020

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Abstract

We consider the nonclassical diffusion equation with hereditary memory

$ u_t-\Delta u_t -\int_0^\infty \kappa(s)\Delta u(t-s)\,{{\rm{d}}} s +f(u) = g $

on a bounded three-dimensional domain. The main feature of the model is that the equation does not contain a term of the form $ -\Delta u $, contributing as an instantaneous damping. Setting the problem in the past history framework, we prove that the related solution semigroup possesses a global attractor of optimal regularity.

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Mathematics Subject Classification: Primary: 35K70, 45K05; Secondary: 35B40, 35B41.

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