Abstract
The nonlinear integro-differential system associated with the penetration of a magnetic field into a substance is considered. The asymptotic behavior as t → ∞ of solutions for two initial-boundary value problems are studied. The problem with non-zero conditions on one side of the lateral boundary is discussed. The problem with homogeneous boundary conditions is studied too. The rates of convergence are given. Results presented show the difference between stabilization characters of solutions of these two cases.
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A. L. Amadori, K.H. Karlsen, C. La Chioma: Non-linear degenerate integro-partial differential evolution equations related to geometric Lévy processes and applications to backward stochastic differential equations. Stochastics Stochastics Rep. 76 (2004), 147–177.
J.M. Chadam, H.M. Yin: An iteration procedure for a class of integrodifferential equations of parabolic type. J. Integral Equations Appl. 2 (1990), 31–47.
B.D. Coleman, M. E. Gurtin: On the stability against shear waves of steady flows of non-linear viscoelastic fluids. J. Fluid Mech. 33 (1968), 165–181.
C.M. Dafermos: An abstract Volterra equation with application to linear viscoelasticity. J. Differ. Equations 7 (1970), 554–569.
C. Dafermos: Stabilizing effects of dissipation. Proc. Int. Conf. Equadiff 82, Würzburg 1982. Lect. Notes Math. Vol. 1017. 1983, pp. 140–147.
C.M. Dafermos, J.A. Nohel: A nonlinear hyperbolic Volterra equation in viscoelasticity. Contributions to analysis and geometry. Suppl. Am. J. Math. (1981), 87–116.
H. Engler: Global smooth solutions for a class of parabolic integrodifferential equations. Trans. Am. Math. Soc. 348 (1996), 267–290.
H. Engler: On some parabolic integro-differential equations: Existence and asymptotics of solutions. Proc. Int. Conf. Equadiff 82, Würzburg 1982. Lect. Notes Math. Vol. 1017. 1983, pp. 161–167.
D.G. Gordeziani, T.A. Jangveladze (Dzhangveladze), T.K. Korshiya: Existence and uniqueness of the solution of certain nonlinear parabolic problems. Differ. Equations 19 (1983), 887–895.
G. Gripenberg: Global existence of solutions of Volterra integrodifferential equations of parabolic type. J. Differ. Equations 102 (1993), 382–390.
G. Gripenberg, S.-O. Londen, O. Staffans: Volterra Integral and Functional Equations. Encyclopedia of Mathematics and Its Applications, Vol. 34. Cambridge University Press, Cambridge, 1990.
M.E. Gurtin, A.C. Pipkin: A general theory of heat conduction with finite wave speeds. Arch. Ration. Mech. Anal. 31 (1968), 113–126.
T.A. Jangvelazde (Dzhangveladze): On the solvability of the first boundary value problem for a nonlinear integro-differential equation of parabolic type. Soobsch. Akad. Nauk Gruz. SSR 114 (1984), 261–264. (In Russian.)
T.A. Jangveladze (Dzhangveladze), Z.V. Kiguradze: Asymptotic behavior of the solution of a nonlinear integro-differential diffusion equation. Differ. Equ. 44 (2008), 538–550.
T.A. Jangveladze (Dzhangveladze), Z.V. Kiguradze: Asymptotics of a solution of a nonlinear system of diffusion of a magnetic field into a substance. Sib. Mat. Zh. 47 (2006), 1058–1070 (In Russian.);, English translation: Sib. Math. J. 47 (2006), 867–878.
T.A. Jangveladze (Dzhangveladze), Z.V. Kiguradze: Estimates of the stabilization rate as t → ∞ of solutions of the nonlinear integro-differential diffusion system. Appl. Math. Inform. Mech. 8 (2003), 1–19.
T.A. Jangveladze (Dzhangvelazde), Z.V. Kiguradze: On the stabilization of solutions of an initial-boundary value problem for a nonlinear integro-differential equation. Differ. Equ. 43 (2007), 854–861;, Translation from Differ. Uravn. 43 (2007), 833–840. (In Russian.)
T.A. Jangveladze (Dzhangvelazde), B.Ya. Lyubimov, T.K. Korshiya: Numerical solution of a class of non-isothermal diffusion problems of an electromagnetic field. Tr. Inst. Prikl. Mat. Im. I.N. Vekua 18 (1986), 5–47. (In Russian.)
J. Kačur: Application of Rothe’s method to evolution integrodifferential equations. J. Reine Angew. Math. 388 (1988), 73–105.
L.D. Landau, E.M. Lifshitz: Electrodynamics of Continuous Media. Pergamon Press, Oxford-London-New York, 1960.
G. Laptev: Mathematical singularities of a problem on the penetration of a magnetic field into a substance. Zh. Vychisl. Mat. Mat. Fiz. 28 (1988), 1332–1345 (In Russian.); English translation:, U.S.S.R. Comput. Math. Math. Phys. 28 (1990), 35–45.
G. Laptev: Quasilinear parabolic equations which contains in coefficients Volterra’s operator. Math. Sbornik 136 (1988), 530–545 (In Russian.);, English translation: Sbornik Math. 64 (1989), 527–542.
J.-L. Lions: Quelques méthodes de résolution des problèmes aux limites non-linéaires. Dunod/Gauthier-Villars, Paris, 1969. (In French.)
N.T. Long, A.P.N. Dinh: Nonlinear parabolic problem associated with the penetration of a magnetic field into a substance. Math. Methods Appl. Sci. 16 (1993), 281–295.
N.T. Long, A.P.N. Dinh: Periodic solutions of a nonlinear parabolic equation associated with the penetration of a magnetic field into a substance. Comput. Math. Appl. 30 (1995), 63–78.
R.C. MacCamy: An integro-differential equation with application in heat flow. Q. Appl. Math. 35 (1977), 1–19.
M. Renardy, W. J. Hrusa, J.A. Nohel: Mathematical Problems in Viscoelasticity. Pitman Monographs and Surveys in Pure and Applied Mathematics, Vol. 35. Longman Scientific & Technical/John Wiley & Sons, Harlow/New York, 1987.
M. Vishik: Über die Lösbarkeit von Randwertaufgaben für quasilineare parabolische Gleichungen höherer Ordnung (On solvability of the boundary value problems for higher order quasilinear parabolic equations). Mat. Sb. N. Ser. 59 (1962), 289–325. (In Russian.)
H.M. Yin: The classical solutions for nonlinear parabolic integrodifferential equations. J. Integral Equations Appl. 1 (1988), 249–263.
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Jangveladze, T.A., Kiguradze, Z.V. Asymptotics for large time of solutions to nonlinear system associated with the penetration of a magnetic field into a substance. Appl Math 55, 471–493 (2010). https://doi.org/10.1007/s10492-010-0019-3
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DOI: https://doi.org/10.1007/s10492-010-0019-3