Global existence and decay estimates for the semilinear heat equation with memory in $\mathbb{R}^{n}$

[1] Y.K. Chang and R. Ponce. Uniform exponential stability and its applications to bounded solutions of integrodifferential equations in Banach spaces, J. Integral Equations Appl. 30 (2018), 347–369.
[2] J. Chen, T. Xiao and J. Liang. Uniform exponential stability of solutions to abstract Volterra equations, J. Evol.
Equ. 4 (2009), no. 9, 661–674.
[3] J. Chen, J. Liang, T. Xiao. Stability of solutions to integro-differential equations in Hilbert spaces, Bull. Belg.
Math. Soc. 18 (2011), 781–792.
[4] V.V. Chepyzhov, E. Mainini and V. Pata, Stability of abstract linear semigroups arising from heat conduction
with memory, Asymptot. Anal. 46 (2006), 251–273.
[5] V.V. Chepyzhov and A. Miranville, On trajectory and global attactors for semilinear heat equations with fadding
memory, Indiana Univ. Math. J. 55 (2006), 119–168.
[6] V.V. Chepyzhov and V. Pata, Some remarks on stability of semigroups arising from linear viscoelasticity, Asymptotic Anal. 46 (2006), 251–273.
[7] B.D. Coleman and M.E. Gurtin, Equipresence and constitutive equations for rigid heat conductors, Z. Angew.
Math. Phys. 18 (1967), 199–208.
[8] C.M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Ration. Mech. Anal. 37 (1970), 297–308
[9] V. Danese, P.G. Geredeli and V. Pata, Exponential attractors for abstract equations with memory and applications
to viscoelasticity, J. Discrete Contin. Dyn. Syst. 35 (2015), 2881–2904.
[10] G. Da Prato and M. Iannelli, Existence and regularity for a class of integro-differential equations of parabolic type,
J. Math. Anal. Appl. 112 (1985), no. 1, 36–55.
[11] G. Da Prato and A. Lunardi, Solvability on the real line of a class of linear Volterra integro-differential equations
of parabolic type, Ann. Mat. Pura Appl. 150 (1988), no. 4, 67–117.
[12] M. Conti, S. Gatti and V. Pata, Decay rates of volterra equations on R
n, Cent. Eur. J. Math. 5 (2007), 720–732.
[13] S. Gatti, A. Miranville, V. Pata and S. Zelik, Attractors for semilinear equations of viscoelasticity with very low
dissipation, Rocky Mountain J. Math. 38 (2008), 1117–1138.
[14] M. Fabrizio and A. Morro, Mathematical problems in linear viscoelasticity, Society for Industrial and Applied
Mathematics, Philadelphia, 1992.
[15] C. Giorgi, M.G. Naso and V. Pata, Exponential stability in linear heat conduction with memory: A semigroup
approach, Comm. Appl. Anal. 5 (2001), 121–134.[16] M. Grasselli and V. Pata, Uniform attractors of nonautonomous dynamical systems with memory, Evolution
equations, semigroups and functional analysis. Birkh¨auser, Basel, 2002, pp. 155–178.
[17] M.E. Gurtin and A.C. Pipkin, A general theory of heat conduction with finite wave speeds, Arch. Rational Mech.
Anal. 31 (1968), 113–126.
[18] Y. Liu and S. Kawashima, Decay property for a plate equation with memory-type dissipation, Kinet. Relat. Models
4 (2011), no. 2, 531–547.
[19] Y. Liu and S. Kawashima, Global Existence and Asymptotic Behavior of Solution for Quasi-linear Dissipation
Plate Equation, Discrete Contin. Dyn. Syst. 29 (2011), 1113–1139.
[20] R. Ikehata and M. Natsume, Energy decay estimates for wave equations with a fractional damping, Differ. Integral
Equ. 25 (2012), no. 9-10, 939–956.
[21] E. Mainini and G. Mola, Exponential and polynomial decay for first order linear Volterra evolution equations,
Quart. Appl. Math. 67 (2009), 93–111.
[22] E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math.
136 (2012), no. 5, 521–573.
[23] J.W. Nunziato, On heat conduction in materials with memory, Quart. Appl. Math. 29 (1971), 187–204.
[24] R.K. Miller, An integrodifferential equation for rigid heat conductors with memory, J. Math. Anal. Appl. 66
(1978), 331–332.
[25] A. Pazy, Semigroups of linear operators and applications to partial differential equations, Springer-Verlag, 1983.
[26] J. Pr¨uss, Evolutionary integral equations and applications, Monographs in Mathematics, vol. 87, Birkh¨auser
Verlag, Basel, 1993.
[27] B. Said-Houari and S.A. Messaoudi, General decay estimates for a Cauchy viscoelastic wave problem, Comm.
Pure Appl. Anal. 13 (2014), no. 4, 1541–1551.
[28] G.F. Webb, An abstract semilinear Volterra integrodifferential equation, Proc. Amer. Math. Soc. 69 (1978), no.
2, 255–260