In this paper, we show the backward uniqueness in time of solutions to nonlinear integro-differential systems with Neumann or Dirichlet boundary conditions. We also discuss reasonable physical interpretations for our conclusions.
@article{bwmeta1.element.doi-10_2478_s11533-010-0042-3,
author = {Yuzhen Bai},
title = {Backward solutions to nonlinear integro-differential systems},
journal = {Open Mathematics},
volume = {8},
year = {2010},
pages = {807-815},
zbl = {1200.35314},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-010-0042-3}
}
Yuzhen Bai. Backward solutions to nonlinear integro-differential systems. Open Mathematics, Tome 8 (2010) pp. 807-815. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-010-0042-3/
[1] Amadori A.L., Nonlinear integro-differential evolution problems arising in option pricing: a viscosity solutions approach, Differential Integral Equations, 2003, 16(7), 787–811 | Zbl 1052.35083
[2] Dafermos C.M., An abstract Volterra equation with application to linear viscoelasticity, J. Differential Equations, 1970, 7, 554–569 http://dx.doi.org/10.1016/0022-0396(70)90101-4[Crossref]
[3] Dzhangveladze T.A., The first boundary value problem for a nonlinear equation of parabolic type, Soviet Phys. Dokl., 1983, 28(4), 323–324 | Zbl 0538.35045
[4] Dzhangveladze T.A., A nonlinear integro-differential equation of parabolic type, Differ. Equ., 1985, 21, 32–36 | Zbl 0595.45022
[5] Dzhangveladze T.A., Kiguradze Z.V., On the stabilization of solutions of an initial-boundary value problem for a nonlinear integro-differential equation, Diff. Equ., 2007, 43(6), 854–861 http://dx.doi.org/10.1134/S0012266107060110[Crossref] | Zbl 1136.78018
[6] Dzhangveladze T.A., Kiguradze Z.V., Asymptotic behavior of the solution of a nonlinear integrodifferential diffusion equation, Diff. Equ., 2008, 44(4), 538–550 http://dx.doi.org/10.1134/S0012266108040083[Crossref] | Zbl 1172.35500
[7] Engler H., On Some Parabolic Integro-Differential Equations: Existence and Asymptotics of Solutions, Lecture Notes in Math., 1017, Springer, Berlin, 1983
[8] Engler H., Global smooth solutions for a class of parabolic integrodifferential equations, Trans. Amer. Math. Soc., 1996, 348(1), 267–290 http://dx.doi.org/10.1090/S0002-9947-96-01472-9[Crossref] | Zbl 0848.45002
[9] Evans L.C., Partial Differential Equations, Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, 1998 | Zbl 0902.35002
[10] Fila M., Pulkkinen A., Backward selfsimilar solutions of supercritical parabolic equations, Appl. Math. Lett., 2009, 22(6), 897–901 http://dx.doi.org/10.1016/j.aml.2008.07.018[Crossref] | Zbl 1173.35547
[11] Gordeziani D.G., Dzhangveladze T.A., Korshiya T.K., Existence and uniqueness of the solution of a class of nonlinear parabolic problems, Diff. Equ., 1983, 19, 887–895 | Zbl 0582.35065
[12] Gripenberg G., Global existence of solutions of Volterra integro-differential equations of parabolic type, J. Differential Equations, 1993, 102(2), 382–390 http://dx.doi.org/10.1006/jdeq.1993.1035[Crossref]
[13] Hoff D., Tsyganov E., Time analyticity and backward uniqueness of weak solutions of the Navier-Stokes equations of multidimensional compressible flow, J. Differential Equations, 2008, 245(10), 3068–3094 http://dx.doi.org/10.1016/j.jde.2008.08.006[Crossref][WoS] | Zbl 05365162
[14] Jangveladze T.A., Kiguradze Z.V., Asymptotics of a solution of a nonlinear system of diffusion of a magnetic field into a substance, Siberian Math. J., 2006, 47(5), 867–878 http://dx.doi.org/10.1007/s11202-006-0095-5[Crossref] | Zbl 1150.45316
[15] Jangveladze T., Kiguradze Z., Neta B., Large time behavior of solutions to a nonlinear integro-differential system, J. Math. Anal. Appl., 2009, 351(1), 382–391 http://dx.doi.org/10.1016/j.jmaa.2008.10.016[Crossref] | Zbl 1172.35330
[16] Lions J.-L., Magenes E., Non-homogeneous Boundary Value Problems and Applications, Vol. 1, Springer, Berlin, 1972 | Zbl 0223.35039
[17] Renardy M., Hrusa W.J., Nohel J.A., Mathematical Problems in Viscoelasticity, Wiley&Sons, New York, 1987 | Zbl 0719.73013