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

Document Type : Research Paper

Authors

1 Laboratory of Mathematical Analysis, Probability and Optimizations, University of Biskra, Po. Box 145 Biskra (07000), Algeria

2 Department of Mathematics, Mohamed Khider University, B.P. 145, 07000, Biskra, Algeria

Abstract

In this paper, we study the initial value problem for a semi-linear heat equation with memory in $n$-dimensional space $\mathbb{R}^{n}$. Under a smallness conditions on the initial data, the global existence and decay estimates of the solutions are established. Furthermore, time decay estimates in higher Sobolev space of the solution are provided. The proof is carried out by means of the point-wise decay estimates of the solution in the Fourier space and a fixed point-contraction mapping argument.

Keywords

[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
Volume 13, Issue 2
July 2022
Pages 2271-2285
  • Receive Date: 20 February 2021
  • Revise Date: 08 March 2021
  • Accept Date: 13 March 2021