April  2020, 19(4): 2035-2050. doi: 10.3934/cpaa.2020090

Nonclassical diffusion with memory lacking instantaneous damping

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

*Corresponding author

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

Received  March 2019 Revised  May 2019 Published  January 2020

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.
Citation: Monica Conti, Filippo Dell'Oro, Vittorino Pata. Nonclassical diffusion with memory lacking instantaneous damping. Communications on Pure & Applied Analysis, 2020, 19 (4) : 2035-2050. doi: 10.3934/cpaa.2020090
References:
[1]

E. C. Aifantis, On the problem of diffusion in solids, Acta Mech., 37 (1980), 265-296.  doi: 10.1007/BF01202949.  Google Scholar

[2]

G. I. BarenblattIu. P. Zheltov and I. N. Kochina, Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks, J. Appl. Math. Mech., 24 (1960), 1286-1303.   Google Scholar

[3]

V. Belleri and V. Pata, Attractors for semilinear strongly damped wave equation on $\mathbb{R}^3$, Discrete Cont. Dyn. Sys., 7 (2001), 719-735.  doi: 10.3934/dcds.2001.7.719.  Google Scholar

[4]

P. J. Chen and M. E. Gurtin, On a theory of heat conduction involving two temperatures, Z. Angew. Math. Phys., 19 (1968), 614-627.   Google Scholar

[5]

V. V. ChepyzhovE. Mainini and V. Pata, Stability of abstract linear semigroups arising from heat conduction with memory, Asymptot. Anal., 50 (2006), 269-291.   Google Scholar

[6]

V. V. Chepyzhov and V. Pata, Some remarks on stability of semigroups arising from linear viscoelasticity, Asymptot. Anal., 46 (2006), 251-273.   Google Scholar

[7]

M. ContiV. DaneseC. Giorgi and V. Pata, A model of viscoelasticity with time-dependent memory kernels, Amer. J. Math., 140 (2018), 349-389.  doi: 10.1353/ajm.2018.0008.  Google Scholar

[8]

M. ContiS. Gatti and V. Pata, Uniform decay properties of linear Volterra integro-differential equations, Math. Models Methods Appl. Sci., 18 (2018), 21-45.  doi: 10.1142/S0218202508002590.  Google Scholar

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M. Conti and E. M. Marchini, Wave equations with memory: the minimal state approach, J. Math. Anal. Appl., 384 (2011), 607-625.  doi: 10.1016/j.jmaa.2011.06.009.  Google Scholar

[10]

M. Conti and E. M. Marchini, A remark on nonclassical diffusion equations with memory, Appl. Math. Optim., 73 (2016), 1-21.  doi: 10.1007/s00245-015-9290-8.  Google Scholar

[11]

M. ContiE. M. Marchini and V. Pata, Semilinear wave equations of viscoelasticity in the minimal state framework, Discrete Contin. Dyn. Syst., 27 (2010), 1535-1552.  doi: 10.3934/dcds.2010.27.1535.  Google Scholar

[12]

M. ContiE. M. Marchini and V. Pata, Nonclassical diffusion with memory, Math. Meth. Appl. Sci., 38 (2015), 948-958.  doi: 10.1002/mma.3120.  Google Scholar

[13]

M. ContiE. M. Marchini and V. Pata, Reaction-diffusion with memory in the minimal state framework, Trans. Amer. Math. Soc., 366 (2014), 4969-4986.  doi: 10.1090/S0002-9947-2013-06097-7.  Google Scholar

[14]

M. Conti and V. Pata, On the regularity of global attractors, Discrete Contin. Dyn. Syst., 25 (2009), 1209-1217.  doi: 10.3934/dcds.2009.25.1209.  Google Scholar

[15]

C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Ration. Mech. Anal., 37 (1970), 554-569.  doi: 10.1007/BF00251609.  Google Scholar

[16]

V. DaneseP. G. Geredeli and V. Pata, Exponential attractors for abstract equations with memory and applications to viscoelasticity, Discrete Contin. Dyn. Syst., 35 (2015), 2881-2904.  doi: 10.3934/dcds.2015.35.2881.  Google Scholar

[17]

M. FabrizioC. Giorgi and V. Pata, A new approach to equations with memory, Arch. Ration. Mech. Anal., 198 (2010), 189-232.  doi: 10.1007/s00205-010-0300-3.  Google Scholar

[18]

M. Grasselli and V. Pata, Uniform attractors of nonautonomous systems with memory, in Evolution Equations, Semigroups and Functional Analysis (A. Lorenzi and B. Ruf, Eds.), pp.155–178, Progr. Nonlinear Differential Equations Appl. no. 50, Birkhäuser, Boston, 2002.  Google Scholar

[19]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, Amer. Math. Soc., Providence, 1988.  Google Scholar

[20]

A. Haraux, Systèmes dynamiques dissipatifs et applications, MassonParis, 1991.  Google Scholar

[21]

J. Jäckle, Heat conduction and relaxation in liquids of high viscosity, Phys. A, 162 (1990), 377-404.   Google Scholar

[22]

A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, in Handbook of Differential Equations: Evolutionary Equations, Vol. 4 (C.M. Dafermos and M. Pokorny, Eds.), Elsevier, Amsterdam, 2008. doi: 10.1016/S1874-5717(08)00003-0.  Google Scholar

[23]

V. Pata, Exponential stability in linear viscoelasticity, Quart. Appl. Math., 64 (2006), 499-513.  doi: 10.1007/s00032-009-0098-3.  Google Scholar

[24]

V. Pata and A. Zucchi, Attractors for a damped hyperbolic equation with linear memory, Adv. Math. Sci. Appl., 11 (2001), 505-529.   Google Scholar

[25]

R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Springer, New York, 1988. doi: 10.1007/978-1-4684-0313-8.  Google Scholar

[26]

X. WangL. Yang and C. Zhong, Attractors for the nonclassical diffusion equation with fading memory, J. Math. Anal. Appl., 362 (2010), 327-337.  doi: 10.1016/j.jmaa.2009.09.029.  Google Scholar

[27]

X. Wang and C. Zhong, Attractors for the non-autonomous nonclassical diffusion equation with fading memory, Nonlinear Anal., 71 (2009), 5733-5746.  doi: 10.1016/j.na.2009.05.001.  Google Scholar

show all references

References:
[1]

E. C. Aifantis, On the problem of diffusion in solids, Acta Mech., 37 (1980), 265-296.  doi: 10.1007/BF01202949.  Google Scholar

[2]

G. I. BarenblattIu. P. Zheltov and I. N. Kochina, Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks, J. Appl. Math. Mech., 24 (1960), 1286-1303.   Google Scholar

[3]

V. Belleri and V. Pata, Attractors for semilinear strongly damped wave equation on $\mathbb{R}^3$, Discrete Cont. Dyn. Sys., 7 (2001), 719-735.  doi: 10.3934/dcds.2001.7.719.  Google Scholar

[4]

P. J. Chen and M. E. Gurtin, On a theory of heat conduction involving two temperatures, Z. Angew. Math. Phys., 19 (1968), 614-627.   Google Scholar

[5]

V. V. ChepyzhovE. Mainini and V. Pata, Stability of abstract linear semigroups arising from heat conduction with memory, Asymptot. Anal., 50 (2006), 269-291.   Google Scholar

[6]

V. V. Chepyzhov and V. Pata, Some remarks on stability of semigroups arising from linear viscoelasticity, Asymptot. Anal., 46 (2006), 251-273.   Google Scholar

[7]

M. ContiV. DaneseC. Giorgi and V. Pata, A model of viscoelasticity with time-dependent memory kernels, Amer. J. Math., 140 (2018), 349-389.  doi: 10.1353/ajm.2018.0008.  Google Scholar

[8]

M. ContiS. Gatti and V. Pata, Uniform decay properties of linear Volterra integro-differential equations, Math. Models Methods Appl. Sci., 18 (2018), 21-45.  doi: 10.1142/S0218202508002590.  Google Scholar

[9]

M. Conti and E. M. Marchini, Wave equations with memory: the minimal state approach, J. Math. Anal. Appl., 384 (2011), 607-625.  doi: 10.1016/j.jmaa.2011.06.009.  Google Scholar

[10]

M. Conti and E. M. Marchini, A remark on nonclassical diffusion equations with memory, Appl. Math. Optim., 73 (2016), 1-21.  doi: 10.1007/s00245-015-9290-8.  Google Scholar

[11]

M. ContiE. M. Marchini and V. Pata, Semilinear wave equations of viscoelasticity in the minimal state framework, Discrete Contin. Dyn. Syst., 27 (2010), 1535-1552.  doi: 10.3934/dcds.2010.27.1535.  Google Scholar

[12]

M. ContiE. M. Marchini and V. Pata, Nonclassical diffusion with memory, Math. Meth. Appl. Sci., 38 (2015), 948-958.  doi: 10.1002/mma.3120.  Google Scholar

[13]

M. ContiE. M. Marchini and V. Pata, Reaction-diffusion with memory in the minimal state framework, Trans. Amer. Math. Soc., 366 (2014), 4969-4986.  doi: 10.1090/S0002-9947-2013-06097-7.  Google Scholar

[14]

M. Conti and V. Pata, On the regularity of global attractors, Discrete Contin. Dyn. Syst., 25 (2009), 1209-1217.  doi: 10.3934/dcds.2009.25.1209.  Google Scholar

[15]

C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Ration. Mech. Anal., 37 (1970), 554-569.  doi: 10.1007/BF00251609.  Google Scholar

[16]

V. DaneseP. G. Geredeli and V. Pata, Exponential attractors for abstract equations with memory and applications to viscoelasticity, Discrete Contin. Dyn. Syst., 35 (2015), 2881-2904.  doi: 10.3934/dcds.2015.35.2881.  Google Scholar

[17]

M. FabrizioC. Giorgi and V. Pata, A new approach to equations with memory, Arch. Ration. Mech. Anal., 198 (2010), 189-232.  doi: 10.1007/s00205-010-0300-3.  Google Scholar

[18]

M. Grasselli and V. Pata, Uniform attractors of nonautonomous systems with memory, in Evolution Equations, Semigroups and Functional Analysis (A. Lorenzi and B. Ruf, Eds.), pp.155–178, Progr. Nonlinear Differential Equations Appl. no. 50, Birkhäuser, Boston, 2002.  Google Scholar

[19]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, Amer. Math. Soc., Providence, 1988.  Google Scholar

[20]

A. Haraux, Systèmes dynamiques dissipatifs et applications, MassonParis, 1991.  Google Scholar

[21]

J. Jäckle, Heat conduction and relaxation in liquids of high viscosity, Phys. A, 162 (1990), 377-404.   Google Scholar

[22]

A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, in Handbook of Differential Equations: Evolutionary Equations, Vol. 4 (C.M. Dafermos and M. Pokorny, Eds.), Elsevier, Amsterdam, 2008. doi: 10.1016/S1874-5717(08)00003-0.  Google Scholar

[23]

V. Pata, Exponential stability in linear viscoelasticity, Quart. Appl. Math., 64 (2006), 499-513.  doi: 10.1007/s00032-009-0098-3.  Google Scholar

[24]

V. Pata and A. Zucchi, Attractors for a damped hyperbolic equation with linear memory, Adv. Math. Sci. Appl., 11 (2001), 505-529.   Google Scholar

[25]

R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Springer, New York, 1988. doi: 10.1007/978-1-4684-0313-8.  Google Scholar

[26]

X. WangL. Yang and C. Zhong, Attractors for the nonclassical diffusion equation with fading memory, J. Math. Anal. Appl., 362 (2010), 327-337.  doi: 10.1016/j.jmaa.2009.09.029.  Google Scholar

[27]

X. Wang and C. Zhong, Attractors for the non-autonomous nonclassical diffusion equation with fading memory, Nonlinear Anal., 71 (2009), 5733-5746.  doi: 10.1016/j.na.2009.05.001.  Google Scholar

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