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On initial-boundary value problem for nonlinear integro-differential equations with variable exponents of nonlinearity

Year 2017, Volume: 5 Issue: 3, 128 - 153, 01.07.2017

Abstract

Some
nonlinear parabolic integro-differential equations with variable exponents of
the nonlinearity are considered. The initial-boundary value problem for these
equations is investigated and the existence theorem for the problem is proved.

References

  • R. A. Adams. Sobolev spaces. Academic Press, New York, San Francisco, London, 1975.
  • S. Antontsev, S. Shmarev. Evolution PDEs with nonstandard growth conditions. Existence, uniqueness, localization, blow-up. Atlantis Studies in Diff. Eq., Vol. 4, Paris: Atlantis Press, 2015.
  • J. P. Aubin. Un theoreme de compacite, Comptes rendus hebdomadaires des seances de l’academie des sciences. 256, No 24 (1963) 5042-5044.
  • T. A. Averina, K. A. Rybakov. New methods of analysis of the Poisson delta-impulse in problem of radiotechnics, J. of Radioelectronics. 1 (2013) 1-20.
  • F. Bernis. Existence results for doubly nonlinear higher order parabolic equations on unbounded domains, Math. Ann. 279 (1988) 373-394.
  • M. Bokalo, V. Dmytriv. Boundary value problems for integro-differential equations in asicotropic spaces, Visnyk (Herald) of Lviv Univ. Series Mech.-Math. 59 (2001) 84-101.
  • T. M. Bokalo, O. M. Buhrii. Doubly nonlinear parabolic equations with variable exponents of nonlinearity, Ukr. Math. J. 63, No 5 (2011) 709-728 (Translated from Ukr. Mat. Z. 63, No 5 (2011) 612-628).
  • H. Brezis. Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer, New York, Dordrecht, Heidelberg, London, 2011.
  • M. Briani, R. Natalini, G. Russo. Implicit-explicit numerical schemes for jump-diffusion processes, Calcolo. 44, No 1 (2007) 33-57.
  • O. M. Buhrii. Parabolic variational inequalities without initial conditions, Ph. D. thesis (Lviv, Ukraine, 2001).
  • O. M. Buhrii. Finiteness of time vanishing of the solution of a nonlinear parabolic variational inequality with variable exponent of nonlinearity, Mat. Studii. 24, No 2 (2005) 167-172.
  • O. M. Buhrii On integration by parts formulaes for special type of exponential functions, Mat. Studii. 45, No 2 (2016) 118-131.
  • O. Buhrii, M. Buhrii. On existence in generalized Sobolev spaces the solution of the initial-boundary value problem for nonlinear integro-differential equations arising from theory of European option, Visnyk (Herald) of Lviv Univ. Series Mech.-Math., 2016 (accepted).
  • J. Byström. Sharp Constants for Some Inequalities Connected to The p-Laplace Operator, Jour. of Ineq. in Pure and Appl. Math. 6, No 2 (2005) Article 56.
  • P. Carr, L. Wu. Time-changed Levy processes and option pricing, J. of Financial Economics. 71 (2004) 113-141.
  • C. la Chioma. Integro-differential problems arising in pricing derivatives in jump-diffusion markets. Ph.D. Thesis. (Roma, 2003-2004).
  • M. Chipot, A. Rougirel. On some class of problems with nonlocal source and boundary flux, Adv. Differential Equations. 6, No 9 (2001) 1025-1048.
  • M. Chipot, N.-H. Chang. On some model diffusion problems with a nonlocal lower order term, Chin. Ann. Math. 24, No 2 (2003) 147-166.
  • M. Chipot, N.-H. Chang. Nonlinear nonlocal evolution problems, Rev. R. Acad. Cien. Serie A. Mat. 97, No 3 (2003) 423-445.
  • S. Cifani, E. R. Jakobsen, K. H. Karlsen. The discontinuous Galerkin method for fractional degenerate convection-diffusion equations, BIT. 51, No 4 (2011) 809-844.
  • S. S. Clift. Linear and non-linear monotone methods for valuing financial options under two-factor, jump-diffusion models, Ph.D. thesis in Computer Science. (Waterloo, Ontario, Canada, 2007).
  • L. Diening, P. Harjulehto, P. Hasto, M. Ruzicka. Lebesgue and Sobolev spaces with variable exponents. Springer, Heidelberg, 2011.
  • Yu. A. Dubinskii. Quasylinear elliptic and parabolic equations of any order, Uspekhi Mat. Nauk. Vol. 23, No 1 (139) (1968) 45-90.
  • N. Dunford, J.T. Schwartz. Linear operators. Part 1: General theory. Izd. Inostran Lit., Moscow, 1962. (translated from: Interscience Publ., New York, London, 1958).
  • L. C. Evans. Partial differential equations. Graduate Studies in Mathematics. Amer. Math. Soc., Providence, RI, 1998.
  • X. L. Fan, D. Zhao. On the spaces L^(p(x)) (Ω) and W^(m,p(x)) (Ω), J. Math. Anal. Appl. 263 (2001) 424-446.
  • H. Gajewski, K. Groger, K. Zacharias. Nonlinear operator equations and operator differential equations. Mir, Moscow, 1978. (translated from: Akademie-Verlag, Berlin, 1974).
  • D. Kinderlehrer, G. Stampacchia. Introduction to variational inequalities and its applications. Mir, Moscow, 1983. (translated from: Academic Press, New York, London, Toronto, Sydney, San Francisco, 1980).
  • A.N. Kolmogorov, S.V. Fomin. Elements of theory of functions and functional analysis. Nauka, Moscow, 1972.
  • O. Kováčik, J. Rákosn13 ̆053'fk. On spaces L^(p(x)) and W^(1,p(x)), Czechoslovak Math. J. 41 (116) (1991) 592-618.
  • S. G. Kou. A jump-diffusion model for option pricing, Management Science. 48 (2002) 1086-1101.
  • O. A. Ladyzhenskaya, N.N. Ural’tseva. Linear and quasilinear elliptic equations, 2th edition. Nauka, Moscow, 1973.
  • J. L. Lions. Some methods of solving of nonlinear boundary value problems. Mir, Moscow, 1972. (translated from: Dunod, Gauthier-Villars, Paris, 1969).
  • R. C. Merton. Option pricing when underlying stock returns are discontinuous, J. of Financial Economics. 3 (1976) 125-144.
  • V. P. Mikhailov. Partial differential equations. Nauka, Moscow, 1976.
  • W. Orlicz. Uber Konjugierte Exponentenfolgen. Studia Mathematica (Lviv) 3 (1931) 200-211.
  • O. T. Panat. Problems for hyperbolic equations and hyperbolic-parabolic systems in generalized Sobolev spaces, Ph.D. thesis (Lviv, Ukraine, 2010).
  • J. P. Pinasco. Blow-up for parabolic and hyperbolic problems with variable exponents, Nonlinear Analysis. 71 (2009) 1094-1099.
  • A. Rougirel. Blow-up rate for parabolic problems with nonlocal source and boundary flux, Electronic J. of Diff. Eq. 2003, No 98 (2003) 1-18.
  • Ph. Souplet. Uniform blow-up profiles and boundary behavior for diffusion equations with nonlocal nonlinear source, J. Diff. Equations. 153 (1999) 374-406.
  • T.P. Timsina. Sensitivities in option pricing models, Ph.D. thesis in Mathematics. (Blacksberg, Virginia, USA, 2007).
  • E.C. Titchmarch. Eigenfunction expansions associated with second-order differential equations. Part II. (Clarendon Press, Oxford, 1958).
Year 2017, Volume: 5 Issue: 3, 128 - 153, 01.07.2017

Abstract

References

  • R. A. Adams. Sobolev spaces. Academic Press, New York, San Francisco, London, 1975.
  • S. Antontsev, S. Shmarev. Evolution PDEs with nonstandard growth conditions. Existence, uniqueness, localization, blow-up. Atlantis Studies in Diff. Eq., Vol. 4, Paris: Atlantis Press, 2015.
  • J. P. Aubin. Un theoreme de compacite, Comptes rendus hebdomadaires des seances de l’academie des sciences. 256, No 24 (1963) 5042-5044.
  • T. A. Averina, K. A. Rybakov. New methods of analysis of the Poisson delta-impulse in problem of radiotechnics, J. of Radioelectronics. 1 (2013) 1-20.
  • F. Bernis. Existence results for doubly nonlinear higher order parabolic equations on unbounded domains, Math. Ann. 279 (1988) 373-394.
  • M. Bokalo, V. Dmytriv. Boundary value problems for integro-differential equations in asicotropic spaces, Visnyk (Herald) of Lviv Univ. Series Mech.-Math. 59 (2001) 84-101.
  • T. M. Bokalo, O. M. Buhrii. Doubly nonlinear parabolic equations with variable exponents of nonlinearity, Ukr. Math. J. 63, No 5 (2011) 709-728 (Translated from Ukr. Mat. Z. 63, No 5 (2011) 612-628).
  • H. Brezis. Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer, New York, Dordrecht, Heidelberg, London, 2011.
  • M. Briani, R. Natalini, G. Russo. Implicit-explicit numerical schemes for jump-diffusion processes, Calcolo. 44, No 1 (2007) 33-57.
  • O. M. Buhrii. Parabolic variational inequalities without initial conditions, Ph. D. thesis (Lviv, Ukraine, 2001).
  • O. M. Buhrii. Finiteness of time vanishing of the solution of a nonlinear parabolic variational inequality with variable exponent of nonlinearity, Mat. Studii. 24, No 2 (2005) 167-172.
  • O. M. Buhrii On integration by parts formulaes for special type of exponential functions, Mat. Studii. 45, No 2 (2016) 118-131.
  • O. Buhrii, M. Buhrii. On existence in generalized Sobolev spaces the solution of the initial-boundary value problem for nonlinear integro-differential equations arising from theory of European option, Visnyk (Herald) of Lviv Univ. Series Mech.-Math., 2016 (accepted).
  • J. Byström. Sharp Constants for Some Inequalities Connected to The p-Laplace Operator, Jour. of Ineq. in Pure and Appl. Math. 6, No 2 (2005) Article 56.
  • P. Carr, L. Wu. Time-changed Levy processes and option pricing, J. of Financial Economics. 71 (2004) 113-141.
  • C. la Chioma. Integro-differential problems arising in pricing derivatives in jump-diffusion markets. Ph.D. Thesis. (Roma, 2003-2004).
  • M. Chipot, A. Rougirel. On some class of problems with nonlocal source and boundary flux, Adv. Differential Equations. 6, No 9 (2001) 1025-1048.
  • M. Chipot, N.-H. Chang. On some model diffusion problems with a nonlocal lower order term, Chin. Ann. Math. 24, No 2 (2003) 147-166.
  • M. Chipot, N.-H. Chang. Nonlinear nonlocal evolution problems, Rev. R. Acad. Cien. Serie A. Mat. 97, No 3 (2003) 423-445.
  • S. Cifani, E. R. Jakobsen, K. H. Karlsen. The discontinuous Galerkin method for fractional degenerate convection-diffusion equations, BIT. 51, No 4 (2011) 809-844.
  • S. S. Clift. Linear and non-linear monotone methods for valuing financial options under two-factor, jump-diffusion models, Ph.D. thesis in Computer Science. (Waterloo, Ontario, Canada, 2007).
  • L. Diening, P. Harjulehto, P. Hasto, M. Ruzicka. Lebesgue and Sobolev spaces with variable exponents. Springer, Heidelberg, 2011.
  • Yu. A. Dubinskii. Quasylinear elliptic and parabolic equations of any order, Uspekhi Mat. Nauk. Vol. 23, No 1 (139) (1968) 45-90.
  • N. Dunford, J.T. Schwartz. Linear operators. Part 1: General theory. Izd. Inostran Lit., Moscow, 1962. (translated from: Interscience Publ., New York, London, 1958).
  • L. C. Evans. Partial differential equations. Graduate Studies in Mathematics. Amer. Math. Soc., Providence, RI, 1998.
  • X. L. Fan, D. Zhao. On the spaces L^(p(x)) (Ω) and W^(m,p(x)) (Ω), J. Math. Anal. Appl. 263 (2001) 424-446.
  • H. Gajewski, K. Groger, K. Zacharias. Nonlinear operator equations and operator differential equations. Mir, Moscow, 1978. (translated from: Akademie-Verlag, Berlin, 1974).
  • D. Kinderlehrer, G. Stampacchia. Introduction to variational inequalities and its applications. Mir, Moscow, 1983. (translated from: Academic Press, New York, London, Toronto, Sydney, San Francisco, 1980).
  • A.N. Kolmogorov, S.V. Fomin. Elements of theory of functions and functional analysis. Nauka, Moscow, 1972.
  • O. Kováčik, J. Rákosn13 ̆053'fk. On spaces L^(p(x)) and W^(1,p(x)), Czechoslovak Math. J. 41 (116) (1991) 592-618.
  • S. G. Kou. A jump-diffusion model for option pricing, Management Science. 48 (2002) 1086-1101.
  • O. A. Ladyzhenskaya, N.N. Ural’tseva. Linear and quasilinear elliptic equations, 2th edition. Nauka, Moscow, 1973.
  • J. L. Lions. Some methods of solving of nonlinear boundary value problems. Mir, Moscow, 1972. (translated from: Dunod, Gauthier-Villars, Paris, 1969).
  • R. C. Merton. Option pricing when underlying stock returns are discontinuous, J. of Financial Economics. 3 (1976) 125-144.
  • V. P. Mikhailov. Partial differential equations. Nauka, Moscow, 1976.
  • W. Orlicz. Uber Konjugierte Exponentenfolgen. Studia Mathematica (Lviv) 3 (1931) 200-211.
  • O. T. Panat. Problems for hyperbolic equations and hyperbolic-parabolic systems in generalized Sobolev spaces, Ph.D. thesis (Lviv, Ukraine, 2010).
  • J. P. Pinasco. Blow-up for parabolic and hyperbolic problems with variable exponents, Nonlinear Analysis. 71 (2009) 1094-1099.
  • A. Rougirel. Blow-up rate for parabolic problems with nonlocal source and boundary flux, Electronic J. of Diff. Eq. 2003, No 98 (2003) 1-18.
  • Ph. Souplet. Uniform blow-up profiles and boundary behavior for diffusion equations with nonlocal nonlinear source, J. Diff. Equations. 153 (1999) 374-406.
  • T.P. Timsina. Sensitivities in option pricing models, Ph.D. thesis in Mathematics. (Blacksberg, Virginia, USA, 2007).
  • E.C. Titchmarch. Eigenfunction expansions associated with second-order differential equations. Part II. (Clarendon Press, Oxford, 1958).
There are 42 citations in total.

Details

Primary Language English
Journal Section Articles
Authors

Oleh Buhrii This is me

Nataliya Buhrii This is me

Publication Date July 1, 2017
Published in Issue Year 2017 Volume: 5 Issue: 3

Cite

APA Buhrii, O., & Buhrii, N. (2017). On initial-boundary value problem for nonlinear integro-differential equations with variable exponents of nonlinearity. New Trends in Mathematical Sciences, 5(3), 128-153.
AMA Buhrii O, Buhrii N. On initial-boundary value problem for nonlinear integro-differential equations with variable exponents of nonlinearity. New Trends in Mathematical Sciences. July 2017;5(3):128-153.
Chicago Buhrii, Oleh, and Nataliya Buhrii. “On Initial-Boundary Value Problem for Nonlinear Integro-Differential Equations With Variable Exponents of Nonlinearity”. New Trends in Mathematical Sciences 5, no. 3 (July 2017): 128-53.
EndNote Buhrii O, Buhrii N (July 1, 2017) On initial-boundary value problem for nonlinear integro-differential equations with variable exponents of nonlinearity. New Trends in Mathematical Sciences 5 3 128–153.
IEEE O. Buhrii and N. Buhrii, “On initial-boundary value problem for nonlinear integro-differential equations with variable exponents of nonlinearity”, New Trends in Mathematical Sciences, vol. 5, no. 3, pp. 128–153, 2017.
ISNAD Buhrii, Oleh - Buhrii, Nataliya. “On Initial-Boundary Value Problem for Nonlinear Integro-Differential Equations With Variable Exponents of Nonlinearity”. New Trends in Mathematical Sciences 5/3 (July 2017), 128-153.
JAMA Buhrii O, Buhrii N. On initial-boundary value problem for nonlinear integro-differential equations with variable exponents of nonlinearity. New Trends in Mathematical Sciences. 2017;5:128–153.
MLA Buhrii, Oleh and Nataliya Buhrii. “On Initial-Boundary Value Problem for Nonlinear Integro-Differential Equations With Variable Exponents of Nonlinearity”. New Trends in Mathematical Sciences, vol. 5, no. 3, 2017, pp. 128-53.
Vancouver Buhrii O, Buhrii N. On initial-boundary value problem for nonlinear integro-differential equations with variable exponents of nonlinearity. New Trends in Mathematical Sciences. 2017;5(3):128-53.