Abstract nonlinear Volterra integrodifferential equations with nonsmooth kernels

Maurizio Grasselli; Alfredo Lorenzi

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (1991)

  • Volume: 2, Issue: 1, page 43-53
  • ISSN: 1120-6330

Abstract

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A Cauchy problem for an abstract nonlinear Volterra integrodifferential equation is considered. Existence and uniqueness results are shown for any given time interval under weak time regularity assumptions on the kernel. Some applications to the heat flow with memory are presented.

How to cite

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Grasselli, Maurizio, and Lorenzi, Alfredo. "Abstract nonlinear Volterra integrodifferential equations with nonsmooth kernels." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 2.1 (1991): 43-53. <http://eudml.org/doc/244193>.

@article{Grasselli1991,
abstract = {A Cauchy problem for an abstract nonlinear Volterra integrodifferential equation is considered. Existence and uniqueness results are shown for any given time interval under weak time regularity assumptions on the kernel. Some applications to the heat flow with memory are presented.},
author = {Grasselli, Maurizio, Lorenzi, Alfredo},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Integrodifferential Volterra equations; Monotone operators; Contraction principle; Heat flow in materials with memory; well-posedness; Cauchy problem; abstract, nonlinear Volterra integrodifferential equations; singular kernel; convolution kernel; integrable singularity; initial-boundary value problem; heat conduction; material with memory},
language = {eng},
month = {3},
number = {1},
pages = {43-53},
publisher = {Accademia Nazionale dei Lincei},
title = {Abstract nonlinear Volterra integrodifferential equations with nonsmooth kernels},
url = {http://eudml.org/doc/244193},
volume = {2},
year = {1991},
}

TY - JOUR
AU - Grasselli, Maurizio
AU - Lorenzi, Alfredo
TI - Abstract nonlinear Volterra integrodifferential equations with nonsmooth kernels
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 1991/3//
PB - Accademia Nazionale dei Lincei
VL - 2
IS - 1
SP - 43
EP - 53
AB - A Cauchy problem for an abstract nonlinear Volterra integrodifferential equation is considered. Existence and uniqueness results are shown for any given time interval under weak time regularity assumptions on the kernel. Some applications to the heat flow with memory are presented.
LA - eng
KW - Integrodifferential Volterra equations; Monotone operators; Contraction principle; Heat flow in materials with memory; well-posedness; Cauchy problem; abstract, nonlinear Volterra integrodifferential equations; singular kernel; convolution kernel; integrable singularity; initial-boundary value problem; heat conduction; material with memory
UR - http://eudml.org/doc/244193
ER -

References

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  1. ACQUISTAPACE, P., Existence and maximal time regularity for linear parabolic integrodifferential equations. J. Int. Equations, 10, 1985, 5-43. Zbl0584.45012MR831233
  2. AIZICOVICI, S., Time-dependent Volterra integrodifferential equations. Ibid., 45-60. Zbl0587.45017MR831234
  3. CANNON, J. R. - LIN, Y., Smooth solutions for an integro-differential equation of parabolic type. Diff. Int. Equations, 2, 1989, 111-121. Zbl0719.45007MR960018
  4. CHEN, G. - GRIMMER, R., Semigroups and integral equations. J. Int. Equations, 2, 1980, 133-154. Zbl0449.45007MR572484
  5. CRANDALL, M. G. - LONDEN, S.-O. - NOHEL, J. A., An abstract nonlinear Volterra integrodifferential equation. J. Math. Anal. Appl, 64, 1978, 701-735. Zbl0395.45023MR500052
  6. DI BLASIO, G., Nonautonomous integrodifferential equations in L p spaces. J. Int. Equations, 10, 1985, 111-121. Zbl0585.45006MR831238
  7. GRIPENBERG, G., Volterra integro-differential equations with accretive nonlinearity. J. Diff. Equations, 60, 1985, 57-79. Zbl0575.45013MR808257DOI10.1016/0022-0396(85)90120-2
  8. HEARD, M. L., An abstract parabolic Volterra integrodifferential equation. SIAM J. Math. Anal., 13, 1982, 81-105. Zbl0477.45008MR641542DOI10.1137/0513006
  9. HEARD, M. L. - RANKIN III, S. M., Weak solutions for a class of parabolic Volterra integrodifferential equations. J. Math. Anal. Appl., 139, 1989, 78-109. Zbl0681.45010MR991928DOI10.1016/0022-247X(89)90231-X
  10. LIONS, J. L., Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod, Paris1969. Zbl0189.40603MR259693
  11. LIONS, J. L. - MAGENES, E., Non-Homogeneous Boundary Value Problems and Applications. Vol. I, Springer Verlag, Berlin1972. Zbl0223.35039MR350177
  12. LONDEN, S.-O. - NOHEL, J. A., Nonlinear Volterra integrodifferential equation occuring in heat flow. J. Int. Equations, 6, 1984, 11-50. Zbl0537.45011MR727934
  13. LUNARDI, A. - SINESTRARI, E., Existence in the large and stability for nonlinear Volterra equations. Ibid., 213-239. Zbl0587.45016MR831245
  14. NOHEL, J. A., Nonlinear Volterra Equations for Heat Flow in Materials with Memory. Integral and Functional Differential Equations, Lecture Notes in Pure and Applied Math., No. 67, Marcel Dekker, New York1981, 3-82. Zbl0465.45017MR617039
  15. NUNZIATO, J. W., On heat conduction in materials with memory. Quart. Appl. Math., 29, 1971, 187-204. Zbl0227.73011MR295683
  16. SINESTRARI, E., An integrodifferential equation arising from the theory of nonlinear heat flow with memory. Nonlinear parabolic equations: qualitative properties of solutions, Pitman Res. Notes in Math., No. 149, Longman, Harlow1987, 207-218. Zbl0624.45016MR901111
  17. WEBB, G. F., An abstract semilinear Volterra integrodifferential equation. Proc. A.M.S., 69, 1978, 255-261. Zbl0388.45012MR467214

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