Continuous dependence of solutions to double dispersive equation with dissipative term

Yıl 2021, Cilt: 50 Sayı: 3, 692 - 698, 07.06.2021

Mesude Elif Uysal , Şevket Gür

https://doi.org/10.15672/hujms.590313

Öz

This work is focused on the properties of solutions to the initial-boundary value problem of the double dispersive-dissipative equation. It is established that the solution depends continuously on the dispersive and dissipative coefficients.

Anahtar Kelimeler

structural stability, double dispersive equations, continuous dependence

Kaynakça

• [1] K.A. Ames and B. Straughan, Non-Standard and improperly posed problems, Mathematics in Science and Engineering Series, Academic Press, New York, 194, 1997.
• [2] N. Bellomo and L. Preziosi, Modelling mathematical methods and scientific computation, CRC, Boca Raton, 1995.
• [3] G. Chen, Y. Wang and S. Wang, Initial boundary value problem of the generalized cubic double dispersion equation, J. Math. Anal. Appl. 299 (2), 563-577, 2004.
• [4] A.O. Çelebi, Ş. Gür and V.K. Kalantarov, Structural stability and decay estimate for marine riser equations, Math. Comput. Model. 54 (11-12), 3182-3188, 2011.
• [5] A.O. Çelebi and V.K. Kalantarov, Decay of solutions and structural stability for the coupled Kuramoto-Sivashinsky-Ginzburg-Landau equations, Appl. Anal. 94 (11), 2342- 2354, 2015.
• [6] H.F. Di and Y.D. Shang, Global existence and asymptotic behavior of solutions for the double dispersive-dissipative wave equation with nonlinear damping and source terms, Bound. Value Probl. 2015, Article No: 29, 2015.
• [7] H.F. Di and Y.D. Shang, Cauchy Problem for a higher order generalized Boussinesqtype equation with hydrodynamical damped term, Appl. Anal. 95 (3), 690-714, 2016.
• [8] H.F. Di and Y.D. Shang, Blow-Up Phenomena for a class of Generalized Double Dispersive Equations, Acta Math. Acad. Sci. 39 (2), 567-579, 2019.
• [9] V.V. Gursky and A.M. Samsonov, Symmetries and exact solutions to a nonlinear doubly dispersive equation with dissipation, Conference: Day on Diffraction 2001, Proceedings, International Seminar, 2001.
• [10] Ş. Gür and I. Güleç, Continuous dependence of solutions to fourth-order nonlinear wave equation, Hacet. J. Math Stat. 45, 367-371, 2016.
• [11] Ş. Gür and M.E. Uysal, Continuous dependence of solutions to strongly damped nonlinear Klein-Gordon Equation, Turkish J. Math. 42, 904-910, 2018.
• [12] C. Lin and L.E. Payne, Continuous dependence on the Soret coefficient for double diffusive convection in Darcy flow, J. Math. Anal. Appl. 342, 311-325, 2008.
• [13] L.E. Payne and B. Straughan, Convergence and continuous Dependence for the Brinkman-Forchheimer equations, Stud. Appl. Math. 102, 419-439, 1999.
• [14] A.M. Samsonov, Nonlinear Strain Waves in Elastic Waveguides, in: Jeffrey A., Engelbrecht J. (eds) Nonlinear Waves in Solids. CISM Courses and Lectures (International Centre for Mechanical Sciences), 341, Springer, Vienna, 1994.
• [15] A.M. Samsonov and E.V. Sokurinskaya, Energy Exchange Between Nonlinear Waves in Elastic Waveguides and External Media, Nonlinear Waves in Active Media, Research Reports in Physics, Springer, Heidelberg, 1989.
• [16] S. Wang and G. Chen, Cauchy problem of the generalized double dispersion equation, Nonlinear Anal. 64 (1) 159-173, 2006.
• [17] M. Waurick, On the continuous dependence on the coefficients of evolutionary equations, Habilitation thesis, TU Dresden, arXiv: 1606.07731, 2016.
• [18] R. Xu, Y. Liu and T. Yu, Global existence of solution for Cauchy problem of multidimensional generalized double dispersion equations, Nonlinear Anal. 71 (10), 4977- 4983, 2009.
• [19] M. Yaman and Ş. Gür, Continuous dependence for the damped nonlinear hyperbolic equation, Math. Comput. Appl. 16 (2), 437-442, 2011.