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Singular Limit of the Generalized Burgers Equation with Absorption

  • Kin Ming Hui
  • Sunghoon KimEmail author
Article
  • 25 Downloads

Abstract

We prove the convergence of the solutions \(u_{m,p}\) of the equation \(u_t+(u^m)_x=-\,u^p\) in \({{\mathbb {R}}}\times (0,\infty )\), \(u(x,0)=u_0(x)\ge 0\) in \({{\mathbb {R}}}\), as \(m\rightarrow \infty \) for any \(p>1\) and \(u_0\in L^1({{\mathbb {R}}})\cap L^{\infty }({{\mathbb {R}}})\) or as \(p\rightarrow \infty \) for any \(m>1\) and \(u_0\in L^{\infty }({{\mathbb {R}}})\). We also show that in general \(\underset{p\rightarrow \infty }{\lim }\underset{m\rightarrow \infty }{\lim }u_{m,p}\ne \underset{m\rightarrow \infty }{\lim }\underset{p\rightarrow \infty }{\lim }u_{m,p}\).

Keywords

Singular limit Generalized Burgers equation Generalized Burgers equation with absorption Kruzkov sense solution Non-commutativity of singular limit 

Mathematics Subject Classification

35B40 35F20 35L02 

Notes

Acknowledgements

Sunghoon Kim was supported by the Research Fund, 2020 of The Catholic University of Korea.

References

  1. 1.
    Bénilan, P., Boccardo, L., Herrero, M.: On the limit of solutions of \(u_t=\varDelta \,u^m\) as \(m\rightarrow \infty \) in some topics in nonlinear PDE’s. In: Bertsch M., et al. (eds.) Proceedings International Conference Torino 1989, Rend. Sem. Mat. Univ. Politec. Torino, 1989. Special Issue, pp. 1–13 (1991)Google Scholar
  2. 2.
    Caffarelli, L.A., Friedman, A.: Asymptotic behaviour of solutions of \(u_t=\varDelta \,u^m\) as \(m\rightarrow \infty \). Indiana Univ. Math. J. 36, 711–723 (1987)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Hui, K.M.: Asymptotic behaviour of solutions of \(u_t=\varDelta \,u^m-u^p\) as \(p\rightarrow \infty \). Nonlinear Anal. TMA 21(3), 191–195 (1993)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Hui, K.M.: Singular limit of solutions of \(u_t=\varDelta \,u^m-A\cdot \nabla (u^q/q)\) as \(q\rightarrow \infty \). Trans. Am. Math. Soc. 347(5), 1687–1712 (1995)Google Scholar
  5. 5.
    Hui, K.M.: Singular limit of solutions of the porous medium equation with absorption. Trans. Am. Math. Soc. 350(11), 4651–4667 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Kružkov, S.N.: First order quasilinear equations in several independent variables. Math. USSR Sbornik 10(2), 217–243 (1970)CrossRefGoogle Scholar
  7. 7.
    Perthame, B.: Some mathematical aspects of tumor growth and therapy. International Congress of Mathematicians Plenary Lecture (2014)Google Scholar
  8. 8.
    Perthame, B., Quiros, F., Vazquez, J.L.: The Hele–Shaw asymptotics for mechanical models of tumor growth. Arch. Ration. Mech. Anal. 212, 93–127 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Perthame, B., Tang, M., Vauchelet, N.: Traveling wave solution of the Hele–Shaw model of tumor growth with nutrient. Math. Models Methods Appl. Sci. 24(13), 2601–2626 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Royden, H.L.: Real Analysis, 2nd edn. Macmillan, New York (1968)zbMATHGoogle Scholar
  11. 11.
    Sacks, P.E.: A singular limit problem for the porous medium equation. J. Math. Anal. Appl. 140, 456–466 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Tao, T.: The high exponent limit \(p\rightarrow \infty \) for the one dimensional nonlinear wave equation. Anal. PDE 2(2), 235–259 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Xu, X.: Asymptotic behaviour of solutions of hyperbolic conservation laws \(u_t+(u^m)_x=0\) as \(m\rightarrow \infty \) with inconsistent initial values. Proc. R. Soc. Edinb. Sect. A 113(1–2), 61–71 (1989)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Institute of MathematicsAcademia SinicaTaipeiTaiwan, R.O.C.
  2. 2.Department of Mathematics, School of Natural SciencesThe Catholic University of KoreaBucheon-siRepublic of Korea

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