In this work, the initial-boundary value problem for one fourth order semilinear hyperbolic equation with memory operator is considered (here the memory operator is under the operator of differentiation with respect to time variable). The asymptotic compactness of semigroup generated by this problem is proved. The existence of a minimal global attractor for this problem is also proved.
[1] N. Kenmochi, A. Visintin, Asymptotic stability for nonlinear evolution problems with hysteresis, Europ. J. Appl. Math. 5 (1994), 39–56.
[2] P. Krejci, Hysterezis and periodic solutions of semilinear and quasilinear wave equations, Math. Z. 193 (1986), 247–264.
[3] P. Krejci, Asymptotic stability of periodic solutions to the wave equation with hysteresis, In: Models of hysteresis (A.Visintin, ed.), Longman, Harlow,
77-90, 1993.
[4] M. Hilpert, On uniqueness for evolution problems with hysteresis, Mathematical Models for Phase Change Problems, (J.F.Rodrigues,ed.), Birkhauser,
Basel, 88, 1989, 377-388.
[5] A. Visintin, Hysteresis and Semigroups, in ”Models of Hysteresis”, A.Visintin, ed. Longman, Harlow, (1993) 192-206.
[6] A. Visinton, Quasilinear hyperbolic equations with hysteresis, Rend. Mat. Acc. Lincei, 15(9)(3-4) (2004), 235-247.
[7] A. Visintin, Differential Models of Hysteresis, Springer, 1993.
[8] J. L. Lions, Some Solution Methods for Nonlinear Boundary Problems, Moscow, Mir, 1972 (in Russian).
[9] S. E. Isayeva, The mixed problem for one semilinear hyperbolic equation with memory, Transactions of NAS of Azerbaijan XXX(1) (2010), 105–112.
[10] R. Bouc, Modele mathematique d’hysteresis et application aux systemes a un degre de liberte, Th´ese, Marseille, 1966.
[11] O. A. Ladyzhenskaya, On the determination of minimal attractors for the Navier-Stokes equations and other partial differential equations, Uspekhi Mat.
Nauk 42(6) (1987), 25–60. English translation: Russian Math. Surveys 42(6) (1987), 27–73.
[12] S.E. Isayeva, The existence of an absorbing set for one mixed problem with memory, Transactions of NAS of Azerbaijan XXXIII(1) (2013), 27–35.
[13] I. Chueshov, I. Lasiecka, Attractors for second-order evolution equations with a nonlinear damping, J. Dynam. Differential Equations 16(2) (2004),
469–512.
[14] I. Lasiecka, A. R. Ruzmaikina, Finite dimensionality and regularity of attractors for 2-D semilinear wave equation with nonlinear dissipation, J. Math.
Anal. Appl. 270 (2002), 16–50.
[15] A. Kh. Khanmamedov, Global attractors for wave equations with nonlinear interior damping and critical exponents, J. Differential Equations 230
(2006), 702–719.
Year 2019,
Volume: 2 Issue: 1, 36 - 41, 20.03.2019
[1] N. Kenmochi, A. Visintin, Asymptotic stability for nonlinear evolution problems with hysteresis, Europ. J. Appl. Math. 5 (1994), 39–56.
[2] P. Krejci, Hysterezis and periodic solutions of semilinear and quasilinear wave equations, Math. Z. 193 (1986), 247–264.
[3] P. Krejci, Asymptotic stability of periodic solutions to the wave equation with hysteresis, In: Models of hysteresis (A.Visintin, ed.), Longman, Harlow,
77-90, 1993.
[4] M. Hilpert, On uniqueness for evolution problems with hysteresis, Mathematical Models for Phase Change Problems, (J.F.Rodrigues,ed.), Birkhauser,
Basel, 88, 1989, 377-388.
[5] A. Visintin, Hysteresis and Semigroups, in ”Models of Hysteresis”, A.Visintin, ed. Longman, Harlow, (1993) 192-206.
[6] A. Visinton, Quasilinear hyperbolic equations with hysteresis, Rend. Mat. Acc. Lincei, 15(9)(3-4) (2004), 235-247.
[7] A. Visintin, Differential Models of Hysteresis, Springer, 1993.
[8] J. L. Lions, Some Solution Methods for Nonlinear Boundary Problems, Moscow, Mir, 1972 (in Russian).
[9] S. E. Isayeva, The mixed problem for one semilinear hyperbolic equation with memory, Transactions of NAS of Azerbaijan XXX(1) (2010), 105–112.
[10] R. Bouc, Modele mathematique d’hysteresis et application aux systemes a un degre de liberte, Th´ese, Marseille, 1966.
[11] O. A. Ladyzhenskaya, On the determination of minimal attractors for the Navier-Stokes equations and other partial differential equations, Uspekhi Mat.
Nauk 42(6) (1987), 25–60. English translation: Russian Math. Surveys 42(6) (1987), 27–73.
[12] S.E. Isayeva, The existence of an absorbing set for one mixed problem with memory, Transactions of NAS of Azerbaijan XXXIII(1) (2013), 27–35.
[13] I. Chueshov, I. Lasiecka, Attractors for second-order evolution equations with a nonlinear damping, J. Dynam. Differential Equations 16(2) (2004),
469–512.
[14] I. Lasiecka, A. R. Ruzmaikina, Finite dimensionality and regularity of attractors for 2-D semilinear wave equation with nonlinear dissipation, J. Math.
Anal. Appl. 270 (2002), 16–50.
[15] A. Kh. Khanmamedov, Global attractors for wave equations with nonlinear interior damping and critical exponents, J. Differential Equations 230
(2006), 702–719.
Isayeva, S. (2019). The Existence of a Global Attractor for one Fourth Order Hyperbolic Equation with Memory Operator. Universal Journal of Mathematics and Applications, 2(1), 36-41. https://doi.org/10.32323/ujma.418626
AMA
Isayeva S. The Existence of a Global Attractor for one Fourth Order Hyperbolic Equation with Memory Operator. Univ. J. Math. Appl. March 2019;2(1):36-41. doi:10.32323/ujma.418626
Chicago
Isayeva, Sevda. “The Existence of a Global Attractor for One Fourth Order Hyperbolic Equation With Memory Operator”. Universal Journal of Mathematics and Applications 2, no. 1 (March 2019): 36-41. https://doi.org/10.32323/ujma.418626.
EndNote
Isayeva S (March 1, 2019) The Existence of a Global Attractor for one Fourth Order Hyperbolic Equation with Memory Operator. Universal Journal of Mathematics and Applications 2 1 36–41.
IEEE
S. Isayeva, “The Existence of a Global Attractor for one Fourth Order Hyperbolic Equation with Memory Operator”, Univ. J. Math. Appl., vol. 2, no. 1, pp. 36–41, 2019, doi: 10.32323/ujma.418626.
ISNAD
Isayeva, Sevda. “The Existence of a Global Attractor for One Fourth Order Hyperbolic Equation With Memory Operator”. Universal Journal of Mathematics and Applications 2/1 (March 2019), 36-41. https://doi.org/10.32323/ujma.418626.
JAMA
Isayeva S. The Existence of a Global Attractor for one Fourth Order Hyperbolic Equation with Memory Operator. Univ. J. Math. Appl. 2019;2:36–41.
MLA
Isayeva, Sevda. “The Existence of a Global Attractor for One Fourth Order Hyperbolic Equation With Memory Operator”. Universal Journal of Mathematics and Applications, vol. 2, no. 1, 2019, pp. 36-41, doi:10.32323/ujma.418626.
Vancouver
Isayeva S. The Existence of a Global Attractor for one Fourth Order Hyperbolic Equation with Memory Operator. Univ. J. Math. Appl. 2019;2(1):36-41.