×

Pseudo almost periodic solutions to integro-differential equations of heat conduction in materials with memory. (English) Zbl 1253.35195

Summary: We investigate the existence of pseudo almost periodic solutions for a class of integro-differential equation with nonlocal initial conditions arising in the study of heat conduction in materials with memory. We first establish some existence theorems for abstract semilinear integro-differential equations with nonlocal initial conditions, and then, we apply our abstract results to the addressed integro-differential equation arising in the study of heat conduction in materials with memory.

MSC:

35R09 Integro-partial differential equations
35B10 Periodic solutions to PDEs
74D99 Materials of strain-rate type and history type, other materials with memory (including elastic materials with viscous damping, various viscoelastic materials)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Gurtin, M. E.; Pipkin, A. C., A general theory of heat conduction with finite wave speeds, Arch. Ration. Mech. Anal., 31, 113-126 (1968) · Zbl 0164.12901
[2] Miller, R. K., An integrodifferential equation for rigid heat conductors with memory, J. Math. Anal. Appl., 66, 313-332 (1978) · Zbl 0391.45012
[3] Grimmer, R., Resolvent operators for integral equations in a Banach space, Trans. Amer. Math. Soc., 273, 333-349 (1982) · Zbl 0493.45015
[4] Liang, J.; Xiao, T. J., Semilinear integrodifferential equations with nonlocal initial conditions, Comput. Math. Appl., 47, 863-875 (2004) · Zbl 1068.45014
[5] Liang, J.; Liu, J. H.; Xiao, T. J., Nonlocal Cauchy problems for nonautonomous evolution equations, Commun. Pure Appl. Anal., 5, 529-535 (2006) · Zbl 1143.34320
[6] Liang, J.; van Casteren, J.; Xiao, T. J., Nonlocal Cauchy problems for semilinear evolution equations, Nonlinear Anal., 50, 173-189 (2002) · Zbl 1009.34052
[7] Xiao, T. J.; Liang, J., Existence of classical solutions to nonautonomous nonlocal parabolic problems, Nonlinear Anal., 63, 225-232 (2005)
[8] Liang, J.; Liu, J. H.; Xiao, T. J., Nonlocal problems for integrodifferential equations, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 15, 815-824 (2008) · Zbl 1163.45010
[9] Lin, Y.; Liu, J. H., Semilinear integrodifferential equations with nonlocal Cauchy problem, Nonlinear Anal., 26, 1023-1033 (1996) · Zbl 0916.45014
[10] Liu, J. H.; Ezzinbi, K., Non-autonomous integrodifferential equations with nonlocal conditions, J. Integral Equations Appl., 15, 79-93 (2003) · Zbl 1044.45002
[11] Aizicovici, S.; McKibben, M., Existence results for a class of abstract nonlocal Cauchy problems, Nonlinear Anal., 39, 649-668 (2000) · Zbl 0954.34055
[12] Aizicovici, S.; Lee, H., Nonlinear nonlocal Cauchy problems in Banach spaces, Appl. Math. Lett., 18, 401-407 (2005) · Zbl 1084.34002
[13] Caicedo, A.; Cuevas, C.; Henríquez, H. R., Asymptotic periodicity for a class of partial integrodifferential equations, ISRN Math. Anal. Volume, 18 (2011), Article ID 537890 · Zbl 1227.45007
[14] dos Santos, J. P.C.; Cuevas, C., Asymptotically almost automorphic solutions of abstract fractional integro-differential neutral equations, Appl. Math. Lett., 23, 960-965 (2010) · Zbl 1198.45014
[15] Diagana, T.; Hernández, E.; dos Santos, J. P.C., Existence of asymptotically almost automorphic solutions to some abstract partial neutral integro-differential equations, Nonlinear Anal., 71, 248-257 (2009) · Zbl 1172.45002
[16] Ding, H. S.; Liang, J.; Xiao, T. J., Asymptotically almost automorphic solutions for some integrodifferential equations with nonlocal initial conditions, J. Math. Anal. Appl., 338, 141-151 (2008) · Zbl 1142.45005
[17] Hernández, E.; dos Santos, J. P.C., Asymptotically almost periodic and almost periodic solutions for a class of partial integrodifferential equations, Electron. J. Differ. Eq., 38, 1-8 (2006) · Zbl 1085.45001
[18] N’Guérékata, G. M., Almost Automorphic Functions and Almost Periodic Functions in Abstract Spaces (2001), Kluwer Academic/Plnum Publishers: Kluwer Academic/Plnum Publishers New York, Berlin, Moscow · Zbl 1001.43001
[19] Zhang, C., Almost Periodic Type Functions and Ergodicity (2003), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht, Boston, London · Zbl 1068.34001
[20] N’Guérékata, G. M.; Pankov, A., Stepanov-like almost automorphic functions and monotone evolution equations, Nonlinear Anal., 68, 2658-2667 (2008) · Zbl 1140.34399
[21] Desch, W.; Grimmer, R.; Schappacher, W., Some considerations for linear integro-differential equations, J. Math. Anal. Appl., 104, 219-234 (1984) · Zbl 0595.45027
[22] Chen, G., Control and stabilization for the wave equation in a bounded domain, SIAM J. Control, 17, 66-81 (1979) · Zbl 0402.93016
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.