Global classical solutions to a kind of mixed initial-boundary value problem for inhomogeneous quasilinear hyperbolic systems

Yong-Fu Yang

Applications of Mathematics (2012)

  • Volume: 57, Issue: 3, page 231-261
  • ISSN: 0862-7940

Abstract

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In this paper, the mixed initial-boundary value problem for inhomogeneous quasilinear strictly hyperbolic systems with nonlinear boundary conditions in the first quadrant { ( t , x ) : t ≥ 0 , x ≥ 0 } is investigated. Under the assumption that the right-hand side satisfies a matching condition and the system is strictly hyperbolic and weakly linearly degenerate, we obtain the global existence and uniqueness of a C 1 solution and its L 1 stability with certain small initial and boundary data.

How to cite

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Yang, Yong-Fu. "Global classical solutions to a kind of mixed initial-boundary value problem for inhomogeneous quasilinear hyperbolic systems." Applications of Mathematics 57.3 (2012): 231-261. <http://eudml.org/doc/247103>.

@article{Yang2012,
abstract = {In this paper, the mixed initial-boundary value problem for inhomogeneous quasilinear strictly hyperbolic systems with nonlinear boundary conditions in the first quadrant $\lbrace (t,x)\colon t \ge 0, x \ge 0\rbrace $ is investigated. Under the assumption that the right-hand side satisfies a matching condition and the system is strictly hyperbolic and weakly linearly degenerate, we obtain the global existence and uniqueness of a $C^1$ solution and its $L^1$ stability with certain small initial and boundary data.},
author = {Yang, Yong-Fu},
journal = {Applications of Mathematics},
keywords = {quasilinear hyperbolic system; mixed initial-boundary value problem; global classical solution; weak linear degeneracy; matching conditon; quasilinear hyperbolic system; mixed initial-boundary value problem; global classical solution; weak linear degeneracy; matching condition},
language = {eng},
number = {3},
pages = {231-261},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Global classical solutions to a kind of mixed initial-boundary value problem for inhomogeneous quasilinear hyperbolic systems},
url = {http://eudml.org/doc/247103},
volume = {57},
year = {2012},
}

TY - JOUR
AU - Yang, Yong-Fu
TI - Global classical solutions to a kind of mixed initial-boundary value problem for inhomogeneous quasilinear hyperbolic systems
JO - Applications of Mathematics
PY - 2012
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 57
IS - 3
SP - 231
EP - 261
AB - In this paper, the mixed initial-boundary value problem for inhomogeneous quasilinear strictly hyperbolic systems with nonlinear boundary conditions in the first quadrant $\lbrace (t,x)\colon t \ge 0, x \ge 0\rbrace $ is investigated. Under the assumption that the right-hand side satisfies a matching condition and the system is strictly hyperbolic and weakly linearly degenerate, we obtain the global existence and uniqueness of a $C^1$ solution and its $L^1$ stability with certain small initial and boundary data.
LA - eng
KW - quasilinear hyperbolic system; mixed initial-boundary value problem; global classical solution; weak linear degeneracy; matching conditon; quasilinear hyperbolic system; mixed initial-boundary value problem; global classical solution; weak linear degeneracy; matching condition
UR - http://eudml.org/doc/247103
ER -

References

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  1. Bressan, A., 10.1512/iumj.1988.37.37021, Indiana Univ. Math. J. 37 (1988), 409-420. (1988) Zbl0632.35041MR0963510DOI10.1512/iumj.1988.37.37021
  2. Bressan, A., Liu, T.-P., Yang, T., 10.1007/s002050050165, Arch. Ration. Mech. Anal. 149 (1999), 1-22. (1999) MR1723032DOI10.1007/s002050050165
  3. Chen, Y. L., Global classical solution of the mixed initial-boundary value problem for a kind of the first order quasilinear hyperbolic system, J. Fudan Univ. Nat. Sci. 45 (2006), 625-631. (2006) MR2273247
  4. Greenberg, J. M., Li, T.-T., 10.1016/0022-0396(84)90135-9, J. Differ. Equations 52 (1984), 66-75. (1984) Zbl0576.35080MR0737964DOI10.1016/0022-0396(84)90135-9
  5. Hörmander, L., The lifespan of classical solutions of nonlinear hyperbolic equations, Lecture Notes Math. 1256 Springer Berlin (1987), 214-280. (1987) Zbl0632.35045MR0897781
  6. John, F., 10.1002/cpa.3160270307, Commun. Pure Appl. Math. 27 (1974), 377-405. (1974) Zbl0302.35064MR0369934DOI10.1002/cpa.3160270307
  7. Kong, D. X., Cauchy problem for first order quasilinear hyperbolic systems, J. Fudan Univ. Nat. Sci. 33 (1994), 705-708. (1994) Zbl0938.35573MR1341024
  8. Li, S. M., Cauchy problem for general first order inhomogeneous quasilinear hyperbolic systems, J. Partial Differ. Equations 15 (2002), 46-68. (2002) Zbl1002.35081MR1892623
  9. Li, T.-T., Global Classical Solutions for Quasilinear Hperbolic Systems, Masson/John Wiley Paris/Chichester (1994). (1994) MR1291392
  10. Li, T.-T., Peng, Y.-J., 10.1016/S0362-546X(02)00123-2, Nonlinear Anal., Theory Methods Appl. 52 (2003), 573-583. (2003) Zbl1027.35065MR1937641DOI10.1016/S0362-546X(02)00123-2
  11. Li, T.-T., Peng, Y.-J., Global C 1 solution to the initial-boundary value problem for diagonal hyperbolic systems with linearly degenerate characteristics, J. Partial Differ. Equations 16 (2003), 8-17. (2003) Zbl1045.35043MR1995402
  12. Li, T.-T., Wang, L. B., 10.3934/dcds.2005.12.59, Discrete Contin. Dyn. Syst. 12 (2005), 59-78. (2005) Zbl1067.35044MR2121249DOI10.3934/dcds.2005.12.59
  13. Li, T.-T., Yu, W.-C., Boundary Value Problems for Quasilinear Hyperbolic Systems, Series V, Duke University, Mathematical Department Durham (1985). (1985) 
  14. Li, T.-T., Zhou, Y., Kong, D.-X., 10.1080/03605309408821055, Commun. Partial Differ. Equations 19 (1994), 1263-1317. (1994) Zbl0810.35054MR1284811DOI10.1080/03605309408821055
  15. Li, T.-T., Zhou, Y., Kong, D.-X., Global classical solutions for general quasilinear hyperbolic systems with decay initial data, Nonlinear Anal., Theory Methods Appl. 28 (1997), 1299-1232. (1997) Zbl0874.35068MR1428653
  16. Liu, T.-P., 10.1016/0022-0396(79)90082-2, J. Differ. Equations 33 (1979), 92-111. (1979) MR0540819DOI10.1016/0022-0396(79)90082-2
  17. Liu, T.-P., Yang, T., 10.1002/(SICI)1097-0312(199912)52:12<1553::AID-CPA3>3.0.CO;2-S, Commun. Pure Appl. Math. 52 (1999), 1553-1586. (1999) Zbl1034.35073MR1711037DOI10.1002/(SICI)1097-0312(199912)52:12<1553::AID-CPA3>3.0.CO;2-S
  18. Qin, T. H., Global smooth solutions of dissipative boundary value problems for first order quasilinear hyperbolic systems, Chin. Ann. Math., Ser. B 6 (1985), 289-298. (1985) Zbl0584.35068MR0842971
  19. Wu, P.-X., Global classical solutions to the Cauchy problem for general first order inhomogeneous quasilinear hyperbolic systems, Chin. Ann. Math., Ser. A 27 (2006), 93-108 Chinese. (2006) Zbl1097.35099MR2208263
  20. Zhou, Y., 10.1142/S0252959904000044, Chin. Ann. Math., Ser. B 25 (2004), 37-56. (2004) Zbl1059.35078MR2033949DOI10.1142/S0252959904000044
  21. Zhou, Y., Yang, Y.-F., 10.1016/j.na.2010.04.057, Nonlinear Anal., Theory Methods Appl. 73 (2010), 1543-1561. (2010) Zbl1195.35205MR2661340DOI10.1016/j.na.2010.04.057

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