Nonlinear Functional Analysis and Applications

Kyungnam University, Department of Mathematics Eduaction (경남대학교 수학교육과)
권호 표지
DOI QR코드

DOI QR Code

LOCAL EXISTENCE AND EXPONENTIAL DECAY OF SOLUTIONS FOR A NONLINEAR PSEUDOPARABOLIC EQUATION WITH VISCOELASTIC TERM

  • Received : 2020.06.08
  • Accepted : 2020.12.15
  • Published : 2021.03.15
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we investigate an initial boundary value problem for a nonlinear pseudoparabolic equation. At first, by applying the Faedo-Galerkin, we prove local existence and uniqueness results. Next, by constructing Lyapunov functional, we establish a sufficient condition to obtain the global existence and exponential decay of weak solutions.

Keywords

References

  1. C.J. Amick, J.L. Bona and M.E. Schonbek, Decay of solutions of some nonlinear wave equations, J. Differential Equs. 81(1) (1989), 1-49. https://doi.org/10.1016/0022-0396(89)90176-9
  2. A. Mahmood, N.A. Khan, C. Fetecau, M. Jamil and Q. Rubbab, Exact analytic solutions for the flow of second grade fluid between two longitudinally oscillating cylinders, J. Prime Research in Math., 5 (2009), 192-204.
  3. S. Asghar, T. Hayat and P. D. Ariel, Unsteady Couette flows in a second grade fluid with variable material properties, Commu. Nonlinear Sci. Numer. Simul., 14(1) (2009), 154-159. https://doi.org/10.1016/j.cnsns.2007.07.016
  4. T. Aziz and F.M. Mah, A note on the solutions of some nonlinear equations arising in third-grade fluid flows: An exact approach, The Scientific World Journal, 2014 (2014), Art. ID 109128, 7 pages.
  5. G. Barenblat, I. Zheltov and I. Kochiva, Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks, J. Appl. Math. Mech., 24(5) (1960), 1286-1303. https://doi.org/10.1016/0021-8928(60)90107-6
  6. T.B. Benjamin, J.L. Bona and J.J. Mahony, Models equation of long waves in nonlinear dispersive systems, Philos. Trans. R. Soc. Lond. Ser. A, 272 (1220) (1972), 47-78. https://doi.org/10.1098/rsta.1972.0032
  7. J.L. Bona and V.A. Dougalis. An initial and boundary value problem for a model equation for propagation of long waves, J. Math. Anal. Appl. 75 (1980), 503-522. https://doi.org/10.1016/0022-247x(80)90098-0
  8. A. Bouziani, Solvability of nonlinear pseudoparabolic equation with a nonlocal boundary condition, Nonlinear Anal., 55 (2003), 883-904. https://doi.org/10.1016/j.na.2003.07.011
  9. A. Bouziani, Initial-boundary value problem for a class of pseudoparabolic equations with integral boundary conditions, J. Math. Anal. Appl., 291(2) (2004), 371-386. https://doi.org/10.1016/S0022-247X(03)00590-0
  10. Y. Cao, J. Yin and C. Wang, Cauchy problems of semilinear pseudoparabolic equations, J. Differential Equ., . 246(12) (2009), 4568-4590. https://doi.org/10.1016/j.jde.2009.03.021
  11. Y. Cao, J.X. Lin and Y.H. Li, One-dimensional viscous diffusion of higher order with gradient dependent potentials and sources, Acta. Math. Sin. 246(12) (2018), 4568-4590.
  12. M.M. Cavalcanti, V.N. Domingos Cavalcanti and J.A. Soriano, On the existence and the uniform decay of a hyperbolic, Southeast Asian Bull. Math., 24 (2000), 183-199. https://doi.org/10.1007/s10012-000-0183-6
  13. M.M. Cavalcanti, V.N. Domingos Cavalcanti and J.A. Soriano, Exponential decay for the solution of semilinear viscoelastic wave equations with localized damping, Electronic J. Diffe. Equ., Vol. 2002(44) (2002), pp. 1-14.
  14. M.M. Cavalcanti, V.N. Domingos Cavalcanti and M.L. Santos, Existence and uniform decay rates of solutions to a degenerate system with memory conditions at the boundary, Appl. Math. Comput., 150 (2004), 439-465. https://doi.org/10.1016/S0096-3003(03)00284-4
  15. D.Q. Dai and Y. Huang, A moment problem for one-dimensional nonlinear pseudoparabolic equation, J. Math. Anal. Appl., 328 (2007), 1057-1067. https://doi.org/10.1016/j.jmaa.2006.06.010
  16. T. Hayat, M. Khan and M. Ayub, Some analytical solutions for second grade fluid flows for cylindrical geometries, Math. Comput. Model., 43(1-2) (2006), 16-29. https://doi.org/10.1016/j.mcm.2005.04.009
  17. T. Hayat, F. Shahzad and M. Ayub, Analytical solution for the steady flow of the third grade fluid in a porous half space, Appl. Math. Model., 31(11) (2007), 2424-2432. https://doi.org/10.1016/j.apm.2006.09.008
  18. J.L. Lions, Quelques methodes de resolution des probl'emes aux limites non-lineaires, Dunod-Gauthier-Villars, Paris, 1969.
  19. N.T. Long and A.P.N. Dinh, On a nonlinear parabolic equation involving Bessel's operator associated with a mixed inhomogeneous condition, J. Comput. Appl. Math., 196(1) (2006), 267-284. https://doi.org/10.1016/j.cam.2005.07.024
  20. V.T.T. Mai, N.A. Triet, L.T.P. Ngoc and N.T. Long, Existence, blow-up and exponential decay for a nonlinear Kirchhoff-Carrier-Love equation with Dirichlet conditions, Nonlinear Funct. Anal. Appl., 25(4) (2020), 617-655 https://doi.org/10.22771/NFAA.2020.25.04.02
  21. L.A. Medeiros and M.M. Miranda, Weak solutions for a nonlinear dispersive equation, J. Math. Anal. Appl,. 59 (1977), 432-441. https://doi.org/10.1016/0022-247x(77)90071-3
  22. S.A. Messaoudi, Blow up and global existence in a nonlinear viscoelastic wave equation, Math. Nachr., 260 (2003), 58-66. https://doi.org/10.1002/mana.200310104
  23. L.T.P. Ngoc, N.V. Y, A.P.N. Dinh and N.T. Long, On a nonlinear heat equation associated with Dirichlet-Robin conditions, Numerical Funct. Anal. Opti., 33(2) (2012), 166-189. https://doi.org/10.1080/01630563.2011.594198
  24. L.T.P. Ngoc, N.V. Y, T. M. Thuyet and N.T. Long, On a nonlinear heat equation with viscoelastic term associate with Robin conditions, Applicable Anal., 96(16) (2017), 2717-2736. https://doi.org/10.1080/00036811.2016.1238461
  25. L.T.P. Ngoc and N.T. Long, Exponential decay and blow-up for a system of nonlinear heat equations containing viscoelastic terms and associated with Robin-Dirichlet conditions, Elect. J. Diff, Equ., 2020(106) (2020), 1-26. https://doi.org/10.1186/s13662-019-2438-0
  26. V. Padron, Effect of aggregation on population recovery modeled by a forward-backward pseudoparabolic equation, Trans. Amer. Math. Soc., 356(7) (2004), 2739-2756. https://doi.org/10.1090/S0002-9947-03-03340-3
  27. J.E. Munoz-Rivera and D. Andrade, Exponential decay of nonlinear wave equation with a viscoelastic boundary condition, Math. Methods Appl. Sci. 23 (2000), 41-61. https://doi.org/10.1002/(SICI)1099-1476(20000110)23:1<41::AID-MMA102>3.0.CO;2-B
  28. M. Sajid and T. Hayat, Series solution for steady flow of a third grade fluid through porous space, Transport in Porous Media, 71(2) (2008), 173-183. https://doi.org/10.1007/s11242-007-9118-3
  29. M.L. Santos, Asymptotic behavior of solutions to wave equations with a memory condition at the boundary, Electr. J. Diff. Equ., 2001(73) (2001), 1-11.
  30. Y.D. Shang and B.L. Guo, On the problem of the existence of global solution for a class of nonlinear convolutional intergro-differential equation of pseudoparabolic type, Acta Math. Appl. Sin. 26(3) (2003), 512-524.
  31. R.E. Showalter and T.W. Ting, Asymptotic behavior of solutions of pseudo-parabolic partial differential equations, Annali di Matematica Pura ed Applicata, 90(4) (1971), 241-258. https://doi.org/10.1007/BF02415050
  32. R.E. Showalter, Existence and representation theorems for a semilinear Sobolev equation in Banach space, SIAM J. Math. Anal., 3 (1972), 527-543. https://doi.org/10.1137/0503051
  33. R.E. Showater, Hilbert space methods for partial differential equations, Electr. J. Diff. Equ. Monograph 01, 1994.
  34. T.W. Ting, Certain non-steady flows of second-order fluids, Arch. Ration. Mech. Anal., 14 (1963), 1-26. https://doi.org/10.1007/BF00250690
  35. L. Zhang, Decay of solution of generalized Benjamin-Bona-Mahony-Burgers equations in n-space dimensions, Nonlinear Anal., 25(12) (1995), 1343-1369. https://doi.org/10.1016/0362-546X(94)00252-D
  36. X. Zhu, F. Li and Y. Li, Global solutions and blow-up solutions to a class pseudoparabolic equations with nonlocal term, Appl. Math. Comput. 329 (2018), 38-51. https://doi.org/10.1016/j.amc.2018.02.003