    November  2021, 20(11): 3959-3974. doi: 10.3934/cpaa.2021139

## Damped Klein-Gordon equation with variable diffusion coefficient

 Department of Mathematics and Physics, University of Wisconsin-Parkside, 900 Wood Road, Kenosha, WI 53141, USA

Received  May 2021 Revised  July 2021 Published  November 2021 Early access  August 2021

Fund Project: The financial support from the 2020 Faculty Summer Research Fellowship from the University of Wisconsin-Parkside

We consider a damped Klein-Gordon equation with a variable diffusion coefficient. This problem is challenging because of the equation's unbounded nonlinearity. First, we study the nonlinearity's continuity properties. Then the existence and the uniqueness of the solutions is established. The main result is the continuity of the solution map on the set of admissible parameters. Its application to the parameter identification problem is considered.

Citation: Qinghua Luo. Damped Klein-Gordon equation with variable diffusion coefficient. Communications on Pure and Applied Analysis, 2021, 20 (11) : 3959-3974. doi: 10.3934/cpaa.2021139
##### References:
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##### References:
  R. Côte and C. Muñoz, Multi-solitons for nonlinear klein-gordon equations, Forum Math. Sigma, 2 (2014), 38pp doi: 10.1017/fms.2014.13.   R. Dautray and J. L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-642-58090-1.   P. G. Drazin and R. S. Johnson, Solitons: an Introduction, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 1989.  doi: 10.1017/CBO9781139172059.   L. C. Evans, Partial differential equations, in Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 1998.  D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, in Classics in Mathematics, Berlin, Springer, 2001.  J. Ginibre and G. Velo, The global Cauchy problem for the nonlinear Klein-Gordon equation. II, Ann. Inst. H. Poincaré Anal. Non Linéaire, 6 (1989), 15-35.  S. Gutman, Fréchet differentiability for a damped sine-Gordon equation, J. Math. Anal. Appl., 360 (2009), 503-517.  doi: 10.1016/j.jmaa.2009.06.074.   S. Gutman, Parabolic Regularization For Sine-Gordon Equation, International Journal of Appl. Math and Mech., 6 (2010), 66-93. J. Ha and S. Gutman, Optimal parameters for a damped sine-Gordon equation, J. Korean Math. Soc., 46 (2009), 1105-1117.  doi: 10.4134/JKMS.2009.46.5.1105.   J. Ha and S. i. Nakagiri, Identification problems for the damped Klein-Gordon equations, J. Math. Anal. Appl., 289 (2004), 77-89.  doi: 10.1016/j.jmaa.2003.08.024.   J. Ha, S. i. Nakagiri and H. Tanabe, Fréchet differentiability of solution mappings for semilinear second order evolution equations, J. Math. Anal. Appl., 346 (2008), 374-383.  doi: 10.1016/j.jmaa.2008.05.038.   S. Ibrahim, M. Majdoub and N. Masmoudi, Global solutions for a semilinear, two-dimensional Klein-Gordon equation with exponential-type nonlinearity, Comm. Pure Appl. Math., 59 (2006), 1639-1658.  doi: 10.1002/cpa.20127.   M. Levi, Beating modes in the Josephson junction, in Chaos in nonlinear dynamical systems (Research Triangle Park, N.C., 1984), SIAM, Philadelphia, PA, 1984, 56–73.  Z. Li and L. Zhao, Asymptotic decomposition for nonlinear damped klein-gordon equations, J. Math. Stud., 53 (2020), 329-352.  doi: 10.4208/jms.v53n3.20.06.   J. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Berlin, Heidelberg, New York, Springer-Verlag, 1971.  J. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Springer-Verlag, 1972.  J. C. H. Simon and E. Taflin, The Cauchy problem for nonlinear Klein-Gordon equations, Commun. Math. Phys., 152 (1993), 433-478.  R. Temam, Infinite-dimensional dynamical systems in mechanics and physics, in Applied Mathematical Sciences, New York, Springer, 1997. doi: 10.1007/978-1-4612-0645-3.   C. T. 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