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A note on blow-up criteria for a class of nonlinear dispersive wave equations with dissipation

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Abstract

In this note, we study the Cauchy problem for a class of nonlinear dispersive wave equation with dissipative term on the real line. We establish a new local-in-space blow-up criterion. Our results improve the corresponding ones in the previous paper.

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References

  1. Novruzov, E., Yazar, B.: On blow-up criteria for a class of nonlinear dispersive wave equations with dissipation. Monatshefte Math. 188, 163–181 (2019)

    Article  MathSciNet  Google Scholar 

  2. Holden, H., Raynaud, X.: Global conservative solutions of the generalized hyperelastic-rod wave equation. J. Differ. Equ. 233(2), 448–484 (2007)

    Article  MathSciNet  Google Scholar 

  3. Tian, C., Yan, W., Zhang, H.: The Cauchy problem for the generalized hyperelastic rod equation. Math. Nachr. 287(17–18), 2116–2137 (2014)

    Article  MathSciNet  Google Scholar 

  4. Brandolese, L., Cortez, M.F.: Blowup issues for a class of nonlinear dispersive wave equations. J. Differ. Equ. 256, 3981–3998 (2014)

    Article  MathSciNet  Google Scholar 

  5. Novruzov, E.: Local-in-space blow-up criteria for a class of nonlinear dispersive wave equations. J. Differ. Equ. 263, 5773–5786 (2017)

    Article  MathSciNet  Google Scholar 

  6. Camassa, R., Holm, D.: An integrable shallow-watere quation with peaked solitons. Phys. Rev. Lett. 71, 1661–1664 (1993)

    Article  MathSciNet  Google Scholar 

  7. Camassa, R., Holm, D., Hyman, J.: A new integrable shallow water equation. Adv. Appl. Mech. 31, 1–33 (1994)

    Article  Google Scholar 

  8. Constantin, A., Kolev, B.: On the geometric approach to the motion of inertial mechanical systems. J. Phys. A 35(32), R51–R79 (2002)

    Article  MathSciNet  Google Scholar 

  9. Constantin, A.: On the scattering problem for the Camassa–Holm equation. Proc. R. Soc. Lond. A 457, 953–970 (2001)

    Article  MathSciNet  Google Scholar 

  10. Constantin, A., Escher, J.: Wave breaking for nonlinear nonlocal shallow water equation. Acta Math. 181, 229–243 (1998)

    Article  MathSciNet  Google Scholar 

  11. Constantin, A.: Existence of permanent and breaking waves for a shallow water equation: a geometric approach. Ann. Inst. Fourier 50, 321–362 (2000)

    Article  MathSciNet  Google Scholar 

  12. Constantin, A., Escher, J.: Well-posedness, global existence and blowup phenomena for a periodic quasi-linear hyperbolic equation. Commun. Pure Appl. Math. 51(5), 475–504 (1998)

    Article  MathSciNet  Google Scholar 

  13. Constantin, A., Strauss, W.A.: Stability of peakons. Commun. Pure Appl. Math. 53, 603–610 (2000)

    Article  MathSciNet  Google Scholar 

  14. Constantin, A.: Finite propagation speed for the Camassa–Holm equation. J. Math. Phys. 46(2), 023506 (2005)

    Article  MathSciNet  Google Scholar 

  15. Bressan, A., Constantin, A.: Global conservative solutions of the Camassa-Holm equation. Arch. Ration. Mech. Anal. 183, 215–239 (2007)

    Article  MathSciNet  Google Scholar 

  16. Bressan, A., Constantin, A.: Global dissipative solutions of the Camassa–Holm equation. Anal. Appl. 5, 1–27 (2007)

    Article  MathSciNet  Google Scholar 

  17. Himonas, A., Misiolek, G., Ponce, G., Zhou, Y.: Persistence properties and unique continuation of solutions of the Camassa–Holm equation. Commun. Math. Phys. 271(2), 511–522 (2007)

    Article  MathSciNet  Google Scholar 

  18. Henry, D.: Compactly supported solutions of the Camassa–Holm equation. J. Nonlinear Math. Phys. 12, 342–347 (2005)

    Article  MathSciNet  Google Scholar 

  19. Misiolek, G.: Classical solutions of the periodic Camassa–Holm equation. Geom. Funct. Anal. 12(5), 1080–1104 (2002)

    Article  MathSciNet  Google Scholar 

  20. Zhou, Y.: Wave breaking for a shallow water equation. Nonlinear Anal. 57(1), 137–152 (2004)

    Article  MathSciNet  Google Scholar 

  21. Dai, H.-H.: Model equations for nonlinear dispersive waves in a compressible Mooney–Rivlin rod. Acta Mech. 127(1–4), 193–207 (1998)

    Article  MathSciNet  Google Scholar 

  22. Guo, Z., Zhou, Y.: Wave breaking and persistence properties for the dispersive rod equation. SIAM J. Math. Anal. 40(6), 2567–2580 (2009)

    Article  MathSciNet  Google Scholar 

  23. Hu, Q., Yin, Z.: Blowup and blowup rate of solutions to a weakly dissipative periodic rod equation. J. Math. Phys. 50, 083503 (2009)

    Article  MathSciNet  Google Scholar 

  24. Zhou, Y.: Local well-posedness and blow-up criteria of solutions for a rod equation. Math. Nachr. 278(14), 1726–1739 (2005)

    Article  MathSciNet  Google Scholar 

  25. Wahlen, E.: On the blow-up of solutions to a nonlinear dispersive rod equation. J. Math. Anal. Appl. 323, 1318–1324 (2006)

    Article  MathSciNet  Google Scholar 

  26. Brandolese, L., Cortez, M.F.: On permanent and breaking waves in hyperelastic rods and rings. J. Funct. Anal. 266, 6954–6987 (2014)

    Article  MathSciNet  Google Scholar 

  27. Brandolese, L.: Local-in-space criteria for blowup in shallow water and dispersive rod equations. Commun. Math. Phys. 330, 401–414 (2014)

    Article  MathSciNet  Google Scholar 

  28. Kato, T.: Quasi-linear equation of evolution, with applications to partial differential equations. In: Spectral Theory and Differential Equations, Lecture Notes in Mathematics, vol. 448. Springer, Berlin, pp. 25–70 (1975)

  29. McKean, H.P.: Breakdown of a shallow water equation. Asian J. Math. 2(4), 867–874 (1998)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

This work was supported by Natural Science Foundation of Hunan Province (No. 2018JJ2272), by the Scientific Research Fund of Hunan Provincial Education Department (Nos. 18C0721, 19B381) and Doctoral Research Fund of Hunan University of Arts and Science.

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Authors and Affiliations

  1. Department of Mathematics and Physics, Hunan University of Arts and Science, Changde, 415000, People’s Republic of China

    Xijun Deng

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  1. Xijun Deng
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Correspondence to Xijun Deng.

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Communicated by Adrian Constantin.

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Deng, X. A note on blow-up criteria for a class of nonlinear dispersive wave equations with dissipation. Monatsh Math 194, 503–512 (2021). https://doi.org/10.1007/s00605-020-01452-1

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