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doi: 10.3934/cpaa.2021139
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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 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 & Applied Analysis, doi: 10.3934/cpaa.2021139
References:
[1]

R. Côte and C. Muñoz, Multi-solitons for nonlinear klein-gordon equations, Forum Math. Sigma, 2 (2014), 38pp doi: 10.1017/fms.2014.13.  Google Scholar

[2]

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.  Google Scholar

[3] P. G. Drazin and R. S. Johnson, Solitons: an Introduction, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 1989.  doi: 10.1017/CBO9781139172059.  Google Scholar
[4]

L. C. Evans, Partial differential equations, in Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 1998.  Google Scholar

[5]

D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, in Classics in Mathematics, Berlin, Springer, 2001.  Google Scholar

[6]

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.   Google Scholar

[7]

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.  Google Scholar

[8]

S. Gutman, Parabolic Regularization For Sine-Gordon Equation, International Journal of Appl. Math and Mech., 6 (2010), 66-93.   Google Scholar

[9]

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.  Google Scholar

[10]

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.  Google Scholar

[11]

J. HaS. 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.  Google Scholar

[12]

S. IbrahimM. 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.  Google Scholar

[13]

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.  Google Scholar

[14]

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.  Google Scholar

[15]

J. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Berlin, Heidelberg, New York, Springer-Verlag, 1971.  Google Scholar

[16]

J. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Springer-Verlag, 1972.  Google Scholar

[17]

J. C. H. Simon and E. Taflin, The Cauchy problem for nonlinear Klein-Gordon equations, Commun. Math. Phys., 152 (1993), 433-478.   Google Scholar

[18]

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.  Google Scholar

[19]

C. T. Wildman, Global Existence and Dispersion of Solutions to Nonlinear Klein-Gordon Equations with Potential, University of California, San Diego, 2014.  Google Scholar

show all references

References:
[1]

R. Côte and C. Muñoz, Multi-solitons for nonlinear klein-gordon equations, Forum Math. Sigma, 2 (2014), 38pp doi: 10.1017/fms.2014.13.  Google Scholar

[2]

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.  Google Scholar

[3] P. G. Drazin and R. S. Johnson, Solitons: an Introduction, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 1989.  doi: 10.1017/CBO9781139172059.  Google Scholar
[4]

L. C. Evans, Partial differential equations, in Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 1998.  Google Scholar

[5]

D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, in Classics in Mathematics, Berlin, Springer, 2001.  Google Scholar

[6]

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.   Google Scholar

[7]

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.  Google Scholar

[8]

S. Gutman, Parabolic Regularization For Sine-Gordon Equation, International Journal of Appl. Math and Mech., 6 (2010), 66-93.   Google Scholar

[9]

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.  Google Scholar

[10]

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.  Google Scholar

[11]

J. HaS. 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.  Google Scholar

[12]

S. IbrahimM. 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.  Google Scholar

[13]

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.  Google Scholar

[14]

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.  Google Scholar

[15]

J. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Berlin, Heidelberg, New York, Springer-Verlag, 1971.  Google Scholar

[16]

J. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Springer-Verlag, 1972.  Google Scholar

[17]

J. C. H. Simon and E. Taflin, The Cauchy problem for nonlinear Klein-Gordon equations, Commun. Math. Phys., 152 (1993), 433-478.   Google Scholar

[18]

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.  Google Scholar

[19]

C. T. Wildman, Global Existence and Dispersion of Solutions to Nonlinear Klein-Gordon Equations with Potential, University of California, San Diego, 2014.  Google Scholar

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