Opuscula Math. 35, no. 6 (2015), 935-956
http://dx.doi.org/10.7494/OpMath.2015.35.6.935

 
Opuscula Mathematica

Existence, uniqueness and estimates of classical solutions to some evolutionary system

Lucjan Sapa

Abstract. The theorem of the local existence, uniqueness and estimates of solutions in Hölder spaces for some nonlinear differential evolutionary system with initial conditions is formulated and proved. This system is composed of one partial hyperbolic second-order equation and an ordinary subsystem with a parameter. In the proof of the theorem we use the Banach fixed-point theorem, the Arzeli-Ascola lemma and the integral form of the differential problem.

Keywords: hyperbolic wave equation, telegraph equation, system of nonlinear equations, existence, uniqueness and estimates of solutions, Hölder space.

Mathematics Subject Classification: 35M31, 35A09, 35A01, 35A02, 35B45.

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  1. L. de Broglie, La mécanique ondulatoire et la structure atomique de la matière et du rayonnement, J. Phys. Rad. 8 (1927) 5, 225-241.
  2. R. Carles, R. Danchin, J.C. Saut, Madelung, Gross-Pitaevskii and Korteveg, Nonlinearity 25 (2012), 2843-2873.
  3. T. Człapiński, On the Cauchy problem for quasilinear hyperbolic differential-functional systems of in the Schauder canonic form, Discuss. Math. 10 (1990), 47-68.
  4. T. Człapiński, On the mixed problem for quasilinear partial differential-functional equations of the first order, Z. Anal. Anwend. 16 (1997), 463-478.
  5. M. Danielewski, The Planck-Kleinert crystal, Z. Naturforsch. 62a (2007), 564-568.
  6. J. Evans, N. Shenk, Solutions to axon equations, Biophys. J. 10 (1970), 1090-1101.
  7. J.W. Evans, Nerve axon equations: I linear approximations, Indiana Univ. Math. J. 21 (1972) 9, 877-885.
  8. J.W. Evans, Nerve axon equations: II stability at rest, Indiana Univ. Math. J. 22 (1972) 1, 75-90.
  9. J.W. Evans, Nerve axon equations: III stability of the nerve impulse, Indiana Univ. Math. J. 22 (1972) 6, 577-593.
  10. J.W. Evans, Nerve axon equations: IV the stable and the unstable impulse, Indiana Univ. Math. J. 24 (1975) 12, 1169-1190.
  11. M. Krupa, B. Sanstede, P. Szmolyan, Fast and slow waves in the FitzHugh-Nagumo equation, J. Differential Equations 133 (1997), 49-97.
  12. Y. Li, Maximum principles and the method of upper and lower solutions for time periodic problems of the telegraph equations, J. Math. Anal. Appl. 327 (2007), 997-1009.
  13. W. Likus, V.A. Vladimirov, Solitary waves in the model of active media, taking into account relaxing effects, to appear in Rep. Math. Phys. (2015).
  14. E. Madelung, Quantentheorie in hydrodynamischer form, Z. Phys. A-Hadron. Nucl. 40 (1927), 322-326.
  15. C.V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum, New York, 1992.
  16. M.H. Protter, H.F. Weinberger, Maximum Principles in Differential Equations, Springer-Verlag, New York, 1984.
  • Lucjan Sapa
  • AGH University of Science and Technology, Faculty of Applied Mathematics, al. Mickiewicza 30, 30-059 Krakow, Poland
  • Communicated by Mirosław Lachowicz.
  • Received: 2014-05-12.
  • Revised: 2015-02-12.
  • Accepted: 2015-02-16.
  • Published online: 2015-06-06.
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Cite this article as:
Lucjan Sapa, Existence, uniqueness and estimates of classical solutions to some evolutionary system, Opuscula Math. 35, no. 6 (2015), 935-956, http://dx.doi.org/10.7494/OpMath.2015.35.6.935

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