Compensated compactness and time-periodic solutions to non-autonomous quasilinear telegraph equations

Eduard Feireisl

Aplikace matematiky (1990)

  • Volume: 35, Issue: 3, page 192-208
  • ISSN: 0862-7940

Abstract

top
In the present paper, the existence of a weak time-periodic solution to the nonlinear telegraph equation U t t + d U t - σ ( x , t , U x ) x + a U = f ( x , t , U x , U t , U ) with the Dirichlet boundary conditions is proved. No “smallness” assumptions are made concerning the function f . The main idea of the proof relies on the compensated compactness theory.

How to cite

top

Feireisl, Eduard. "Compensated compactness and time-periodic solutions to non-autonomous quasilinear telegraph equations." Aplikace matematiky 35.3 (1990): 192-208. <http://eudml.org/doc/15624>.

@article{Feireisl1990,
abstract = {In the present paper, the existence of a weak time-periodic solution to the nonlinear telegraph equation $U_\{tt\}+dU_t-\sigma (x,t,U_x)_x+aU=f(x,t,U_x,U_t,U)$ with the Dirichlet boundary conditions is proved. No “smallness” assumptions are made concerning the function $f$. The main idea of the proof relies on the compensated compactness theory.},
author = {Feireisl, Eduard},
journal = {Aplikace matematiky},
keywords = {telegraph equation; compensated compactness; vanishing viscosity method; nonlinear telegraph equation; compensated compactness theory; vanishing viscosity method},
language = {eng},
number = {3},
pages = {192-208},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Compensated compactness and time-periodic solutions to non-autonomous quasilinear telegraph equations},
url = {http://eudml.org/doc/15624},
volume = {35},
year = {1990},
}

TY - JOUR
AU - Feireisl, Eduard
TI - Compensated compactness and time-periodic solutions to non-autonomous quasilinear telegraph equations
JO - Aplikace matematiky
PY - 1990
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 35
IS - 3
SP - 192
EP - 208
AB - In the present paper, the existence of a weak time-periodic solution to the nonlinear telegraph equation $U_{tt}+dU_t-\sigma (x,t,U_x)_x+aU=f(x,t,U_x,U_t,U)$ with the Dirichlet boundary conditions is proved. No “smallness” assumptions are made concerning the function $f$. The main idea of the proof relies on the compensated compactness theory.
LA - eng
KW - telegraph equation; compensated compactness; vanishing viscosity method; nonlinear telegraph equation; compensated compactness theory; vanishing viscosity method
UR - http://eudml.org/doc/15624
ER -

References

top
  1. H. Amann, 10.1016/0022-247X(78)90192-0, J. Math. Anal. Appl. 65 (1978), 432-467. (1978) Zbl0387.35038MR0506318DOI10.1016/0022-247X(78)90192-0
  2. H. Amann, Periodic solutions of semi-linear parabolic equations, Nonlinear Analysis: A collection of papers in honor of Erich Rothe, Academic Press, New York (1978), 1 - 29. (1978) MR0499089
  3. K. N. Chueh C. C. Conley J. A. Smoller, 10.1512/iumj.1977.26.26029, Indiana Univ. Math. J. 26 (1977), 373 - 392. (1977) MR0430536DOI10.1512/iumj.1977.26.26029
  4. W. Craig, A bifurcation theory for periodic solutions of nonlinear dissipative hyperbolic equations, Ann. Sci. Norm Sup. Pisa Ser. IV- Vol. 10 (1983), 125-167. (1983) Zbl0518.35057MR0713113
  5. R. J. DiPerna, 10.1090/S0002-9947-1985-0808729-4, Trans. Amer. Math. Soc. 292 (2) (1985), 383 - 420. (1985) MR0808729DOI10.1090/S0002-9947-1985-0808729-4
  6. R. J. DiPerna, 10.1007/BF00251724, Arch. Rational. Mech. Anal. 82 (1983) 27-70. (1983) Zbl0519.35054MR0684413DOI10.1007/BF00251724
  7. E. Feireisl, Time-dependent invariant regions for parabolic systems related to one-dimensional nonlinear elasticity, Apl. mat. 35 (1990), 184-191. (1990) Zbl0709.73013MR1052739
  8. D. Henry, 10.1007/BFb0089647, Lecture Notes in Math. 840, Springer-Verlag (1981). (1981) Zbl0456.35001MR0610244DOI10.1007/BFb0089647
  9. P. Krejčí, Hard implicit function theorem and small periodic solutions to partial differential equations, Comment. Math. Univ. Carolinae 25 (1984), 519-536. (1984) MR0775567
  10. A. Matsumura, 10.2977/prims/1195189813, Publ. RIMS Kyoto Univ. 13, (1977), 349-379. (1977) Zbl0371.35030MR0470507DOI10.2977/prims/1195189813
  11. A. Milani, Global existence for quasi-linear dissipative wave equations with large data and small parameter, Math. Z. 198 (1988), 291 - 297. (198) MR0939542
  12. A. Milani, 10.1007/BF01776855, Ann. Mat. Рurа Appl. 140 (4) (1985), 331-344. (1985) MR0807643DOI10.1007/BF01776855
  13. T. Nishida, Nonlinear hyperbolic equations and related topics in fluid dynamics, Publications Mathématiques D'Orsay 78.02, Univ. Paris Sud (1978). (1978) Zbl0392.76065MR0481578
  14. H. Petzeltová, Applications of Moser's method to a certain type of evolution equations, Czechoslovak Math. J. 33 (1983), 427-434. (1983) MR0718925
  15. H. Petzeltová M. Štědrý, Time periodic solutions of telegraph equations in n spatial variables, Časopis Pěst. Mat. 109 (1984), 60-73. (1984) MR0741209
  16. P. H. Rabinowitz, 10.1002/cpa.3160220103, Comm. Pure Appl. Math. 22 (1969), 15-39. (1969) Zbl0157.17301MR0236504DOI10.1002/cpa.3160220103
  17. M. Rascle, Un résultat de "compacité par compensation à coefficients variables". Application à l'elasticitě non linéaire, C. R. Acad. Sci. Paris 302 Sér. I 8 (1986), 311 - 314. (1986) Zbl0606.35054MR0838582
  18. D. Serre, La compacité par compensation pour lour les systemes hyperboliques non linéaires de deux équations a une dimension d'espace, J. Math. Pures et Appl. 65 (1986), 423 - 468. (1986) MR0881690
  19. M. Slemrod, Damped conservation laws in continuum mechanics, Nonlinear Analysis and Mechanics Vol. III, Pitman New York (1978), 135-173. (1978) MR0539691
  20. M. Štědrý, Small time-periodic solutions to fully nonlinear telegraph equations in more spatial dimensions, (to appear). MR0995505
  21. L. Tartar, Compensated compactness and applications to partial differential equations, Research Notes in Math. 39, Pitman Press (1975), 136-211. (1975) MR0584398
  22. O. Vejvoda, al., Partial differential equations: Time periodic solutions, Martinus Nijhoff Publ. (1982). (1982) Zbl0501.35001

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.