×

zbMATH — the first resource for mathematics

Existence and sharp regularity results for linear parabolic non- autonomous integro-differential equations. (English) Zbl 0603.45019
The authors solve the linear abstract integro-differential equation of parabolic type \[ du(t)/dt-A(t)u(t)- \int^{t}_{0}B(t,s)u(s)ds=f(t),\quad t\in [0,T],\quad u(0)=u_ 0 \] in a Banach space. It is assumed that the operator A(t) which is not necessarily densely defined generates an analytic semigroup for each fixed t and B(t,s) is an operator whose domain contains that of A(s) for each \(0\leq s<t\leq T.\)
Based on their theory of parabolic evolution equations the authors establish the existence, uniqueness and maximal regularity of the strict solution for any Hölder continuous inhomogeneous term and initial value satisfying a certain compatibility condition.
The main result is applied to second order parabolic integro-partial differential equations in which B(t,s) is a differential operator of second order in case of one space variable and one of first order in case of several space variables. The theory of interpolation spaces by the authors as well as some of other authors such as Da Prato, Grisvard, Lunardi, Sinestrari also plays an important role.
Reviewer: H.Tanabe

MSC:
45N05 Abstract integral equations, integral equations in abstract spaces
45K05 Integro-partial differential equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] P. Acquistapace and B. Terreni,Some existence and regularity results for abstract non-autonomous parabolic equations, J. Math. Anal. Appl.99(1984), 9–61. · Zbl 0555.34051 · doi:10.1016/0022-247X(84)90234-8
[2] P. Acquistapace and B. Terreni,On the abstract non-autonomous Cauchy problem in the case of constant domains, Ann. Mat. Pura Appl. (4)140 (1985), 1–55. · Zbl 0579.34001 · doi:10.1007/BF01776844
[3] P. Acquistapace and B. Terreni,Maximal space regularity for abstract non-autonomous parabolic equations, J. Funct. Anal.60(1985), 168–210. · Zbl 0563.47028 · doi:10.1016/0022-1236(85)90050-3
[4] P. Acquistapace and B. Terreni,An approach to Ito linear equation in Hilbert Spaces by approximation of white noise by coloured noise, Stoch. Anal. Appl.2(1984), 131–186. · Zbl 0547.60066 · doi:10.1080/07362998408809031
[5] P. Acquistapace and B. Terreni,Characterization of Hölder and Zygmund classes as interpolation spaces, preprint, Dipartimento di Matematica, Univ. di Pisa no. 61 (1984). · Zbl 0555.34051
[6] A. Belleni Morante,An integro-differential equation arising from the theory of heat conduction in rigid material with memory, Boll. Un. Mat. Ital. (5)15 B (1978), 470–482. · Zbl 0394.45006
[7] R. W. Carr and K. B. Hannsgen,A non-homogeneous integro-differential equation in Hilbert space, SIAM J. Math. Anal.10(1979), 961–984. · Zbl 0411.45013 · doi:10.1137/0510089
[8] R. W. Carr and K. B. Hannsgen,Resolvent formulas for a Volterra equation in Hilbert space, SIAM J. Math. Anal.13 (1982), 459–483. · Zbl 0501.45015 · doi:10.1137/0513032
[9] B. D. Coleman and M. E. Gurtin,Equipresence and constitutive equations for rigid heat conductors, Z. Angew. Math. Phys.18(1967), 199–208. · doi:10.1007/BF01596912
[10] G. Da Prato and P. Grisvard,Sommes d’opérateurs linéaires et équations différentielles opérationnelles, J. Math. Pures Appl.54(1975), 305–387. · Zbl 0315.47009
[11] G. Da Prato and P. Grisvard,Equations d’evolution abstraites non linéaires de type parabolique, Ann. Mat. Pura Appl. (4)120 (1979), 329–396. · Zbl 0471.35036 · doi:10.1007/BF02411952
[12] G. Da Prato and M. Iannelli,Linear integro-differential equations in Banach spaces, Rend. Sem. Mat. Univ. Padova62 (1980), 207–219. · Zbl 0451.45014
[13] G. Da Prato and M. Iannelli,Existence and regularity for a class of integro-differential equations of parabolic type, J. Math. Anal. Appl., to appear. · Zbl 0583.45009
[14] G. Da Prato, M. Iannelli and E. Sinestrari,Temporal regularity for a class of integro-differential equations in Banach spaces, Boll. Un. Mat. Ital. An. Funz. Appl. (6)2C (1983), 171–185. · Zbl 0527.45005
[15] W. E. Fitzgibbon,Semilinear integro-differential equations in Banach space, Nonlinear Anal.4(1980), 745–760. · Zbl 0442.45014 · doi:10.1016/0362-546X(80)90075-9
[16] A. Friedman,Partial Differential Equations, Holt, Rinehart & Winston, New York/Chicago/San Francisco, 1969. · Zbl 0224.35002
[17] A. Friedman,Monotonicity of solutions of Volterra integral equations in Banach space, Trans. Amer. Math. Soc.138 (1969), 129–148. · Zbl 0182.15201 · doi:10.1090/S0002-9947-1969-0242024-0
[18] A. Friedman and M. Shinbrot,Volterra integral equations in Banach space, Trans. Amer. Math. Soc.126 (1967), 131–179. · Zbl 0147.12302 · doi:10.1090/S0002-9947-1967-0206754-7
[19] R. Grimmer and F. Kappel,Series expansions for resolvents of Volterra integro-differential equations in Banach space, SIAM J. Math. Anal.15 (1984), 595–604. · Zbl 0538.45012 · doi:10.1137/0515045
[20] R. Grimmer and A. J. Pritchard,Analytic resolvent operators for integral equations in Banach space, J. Differ. Equ.50 (1983), 234–259. · Zbl 0519.45011 · doi:10.1016/0022-0396(83)90076-1
[21] M. E. Gurtin,On the thermodynamics of materials with memory, Arch. Rat. Mach. Anal.28 (1968), 40–50. · Zbl 0169.28002 · doi:10.1007/BF00281562
[22] M. L. Heard,An abstract parabolic Volterra integro-differential equation, SIAM J. Math. Anal.13 (1982), 81–105. · Zbl 0477.45008 · doi:10.1137/0513006
[23] T. Kato and H. Tanabe,On the abstract evolution equation, Osaka Math. J.14 (1962), 107–133. · Zbl 0106.09302
[24] A. Lunardi,Interpolation spaces between domains of elliptic operators and spaces of continuous functions with applications to nonlinear parabolic equations, Math. Nachr.121 (1985), 295–318. · Zbl 0568.47035 · doi:10.1002/mana.19851210120
[25] A. Lunardi,Abstract quasilinear parabolic equations, Math. Ann.267 (1984), 395–415. · Zbl 0547.35054 · doi:10.1007/BF01456097
[26] A. Lunardi and E. Sinestrari,C \(\alpha\)-regularity for non-autonomous linear integro-differential equations of parabolic type, J. Differ. Equ., to appear. · Zbl 0596.45019
[27] R. K. Miller,Linear Volterra integro-differential equations as semigroups, Funkcialaj Ekvacioj17 (1974), 39–55. · Zbl 0288.45004
[28] R. K. Miller,Volterra integral equations in a Banach space, Funkcialaj Ekvacioj18 (1975), 163–194. · Zbl 0326.45007
[29] R. K. Miller,An integro-differential equation for rigid heat conductors with memory, J. Math. Anal. Appl.66 (1978), 313–332. · Zbl 0391.45012 · doi:10.1016/0022-247X(78)90234-2
[30] J. Prüss,On resolvent operators for linear integro-differential equations of Volterra type, J. Int. Equ.5 (1983), 211–236.
[31] E. Sinestrari,On the abstract Cauchy problem in spaces of continuous functions, J. Math. Anal. Appl.107 (1985), 16–66. · Zbl 0589.47042 · doi:10.1016/0022-247X(85)90353-1
[32] E. Sinestrari,Continuous interpolations paces and spatial regularity in nonlinear Volterra integro-differential equations, J. Int. Equ.5 (1983), 287–308. · Zbl 0519.45013
[33] H. B. Stewart,Generation of analytic semigroups by strongly elliptic operators under general boundary conditions, Trans. Amer. Math. Soc.259 (1980), 299–310. · Zbl 0451.35033 · doi:10.1090/S0002-9947-1980-0561838-5
[34] H. Tanabe,Note on singular perturbation for abstract differential equations, Osaka J. Math.1 (1964), 239–252. · Zbl 0135.37101
[35] H. Tanabe,Note on nonlinear Volterra integral equations in Hilbert space, Proc. Japan Acad.56 (1980), 9–11. · Zbl 0457.45008 · doi:10.3792/pjaa.56.9
[36] H. Tanabe,Linear Volterra integral equations of parabolic type, Hokkaido Math. J.12 (1983), 265–275. · Zbl 0523.45007
[37] G. F. Webb,An abstract semilinear Volterra integro-differential equation, Proc. Amer. Math. Soc.69 (1978), 255–260. · Zbl 0388.45012 · doi:10.1090/S0002-9939-1978-0467214-4
[38] G. F. Webb,Abstract Volterra integro-differential equations and a class of reaction diffusion equations, inVolterra Equations, Proceedings of the Helsinki Symposium on integral equations, Otaniemi Finland, Lect. Notes in Math. no. 737, Springer-Verlag, Berlin, 1979. · Zbl 0428.45008
[39] A. Yagi,On the abstract linear evolution equations in Banach spaces, J. Math. Soc. Japan28 (1976), 290–303. · Zbl 0318.34068 · doi:10.2969/jmsj/02820290
[40] A. Yagi,On the abstract evolution equation of parabolic type, Osaka J. Math.14 (1977), 557–568. · Zbl 0371.47037
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.