Continuous dependence of solutions to the strongly damped nonlinear Klein–Gordon equation

Continuous dependence of solutions to the strongly damped nonlinear Klein–Gordon equation

This article is devoted to the study of the initial-boundary value problem for the strongly damped nonlinearKlein–Gordon equation. It is proved that the solution depends continuously on changes in the damping terms, diffusion,mass, and nonlinearity effect term in the $H^1$ norm.

___

  • [1] Ames KA, Straughan B. Non-Standard and Improperly Posed Problems. Mathematics in Science and Engineering Series. New York, NY, USA: Academic Press, 1997.
  • [2] Avrin JD. Convergence properties of the strongly damped nonlinear KleinGordon equation. J Differ Equations 1987; 67: 243-255.
  • [3] Ball JM. Finite time blow-up in nonlinear problems. In: Crandall MG, editor. Nonlinear Evolution Equations. New York, NY, USA: Academic Press, 1978, pp. 189-205.
  • [4] Bellomo N, Preziosi L. Modelling Mathematical Methods and Scientific Computation. Boca Raton, FL, USA: CRC Press, 1995.
  • [5] Cazenave T. Uniform estimates for solutions of nonlinear Klein-Gordon equations. J Funct Anal 1985; 1: 36-55.
  • [6] Cazenave T, Haraux A. An Introduction to Semilinear Evolution Equations. Oxford Lecture Series in Mathematics and Its Applications. Oxford, UK: Oxford University Press, 1998.
  • [7] Çelebi AO, G¨ur S¸, Kalantarov VK. Structural stability and decay estimate for marine riser equations. Math Comput Model 2011; 11-12: 3182-3188.
  • [8] Gao P, Guo BL. The time-periodic solution for a 2D dissipative Klein–Gordon equation. J Math Anal Appl 2004; 296: 286-294.
  • [9] Ginibre J, Velo G. The global Cauchy problem for the nonlinear Klein-Gordon equation. Ann I H Poincare-An 1989; 6: 15-35.
  • [10] Güleç İ, Gür Ş. Continuous dependence of solutions to fourth-order nonlinear wave equation. Hacet J Math Stat 2016; 45: 367-371.
  • [11] Ha TG, Park JY. Global existence and uniform decay of a damped Klein-Gordon equation in a noncylindrical domain. Nonlinear Anal 2011; 74: 577-584.
  • [12] Klainerman S. Global existence of small amplitude solutions to nonlinear Klein-Gordon equations in four space-time dimensions. Commun Pur Appl Math 1985; 38: 631-641.
  • [13] Lions JL. Quelques methodes de resolution des problemes aux limites non lineaires. Paris, France: Dunod, 1969 (in French).
  • [14] Nakao M. Energy decay to the Cauchy problem of nonlinear Klein-Gordon equations with a sublinear dissipative term. Adv Math Sci Appl 2009; 19: 479-501.
  • [15] Pecher H. L p -Absch¨atzungen und klassiche L¨osungen f¨ur nichtlineare Wellengleichungen. I. Math Z 1976; 150: 159-183 (in German).
  • [16] Polat N, Taskesen H. On the existence of global solutions for a nonlinear Klein-Gordon equation, Filomat 2014; 28: 1073-1079.
  • [17] Strauss WA. Nonlinear scattering theory at low energy. J Funct Anal 1981; 41: 110-133.
  • [18] Strauss WA. Nonlinear Wave Equations. In: CBMS Regional Conference Series in Mathematics. Published for the Conference Board of the Mathematical Sciences. Providence, RI, USA: AMS, 1989, p. 73.
  • [19] Xu RZ. Global existence, blow up and asymptotic behaviour of solutions for nonlinear Klein-Gordon equation with dissipative term. Math Method Appl Sci 2010; 33: 831-844.
  • [20] Xu R, Ding Y. Global solutions and finite time blow up for damped Klein-Gordon equation. Acta Math Sci 2013; 33B: 643-652.