대한수학회지 (Journal of the Korean Mathematical Society)

  • 제47권4호
  • /
  • Pages.789-804
  • /
  • 2010
  • /
  • 0304-9914(pISSN)
  • /
  • 2234-3008(eISSN)
대한수학회 (Korean Mathematical Society)
권호 표지
DOI QR코드

DOI QR Code

A VERY SINGULAR SOLUTION OF A DOUBLY DEGENERATE PARABOLIC EQUATION WITH NONLINEAR CONVECTION

  • Fang, Zhong Bo (SCHOOL OF MATHEMATICAL SCIENCES OCEAN UNIVERSITY OF CHINA)
  • 투고 : 2008.09.30
  • 발행 : 2010.07.01
PDF 다운로드
⟨ 이전 논문 다음 논문 ⟩

초록

We here investigate an existence and uniqueness of the nontrivial, nonnegative solution of a nonlinear ordinary differential equation: $$[\mid(w^m)]'\mid^{p-2}(w^m)']'\;+\;{\beta}rw'\;+\;{\alpha}w\;+\;(w^q)'\;=\;0$$ satisfying a specific decay rate: $lim_{r\rightarrow\infty}\;r^{\alpha/\beta}w(r)$ = 0 with $\alpha$ := (p - 1)/[pd-(m+1)(p-1)] and $\beta$:= [q-m(p-1)]/[pd-(m+1)(p-1)]. Here m(p-1) > 1 and m(p - 1) < q < (m+1)(p-1). Such a solution arises naturally when we study a very singular solution for a doubly degenerate equation with nonlinear convection: $$u_t\;=\;[\mid(u^m)_x\mid^{p-2}(u^m)_x]_x\;+\;(u^q)x$$ defined on the half line.

키워드

참고문헌

  1. J. S. Baek, M. Kwak, and K. Yu, Uniqueness of the very singular solution of a degenerate parabolic equation, Nonlinear Anal. 45 (2001), no. 1, Ser. A: Theory Methods, 123-135. https://doi.org/10.1016/S0362-546X(99)00334-X
  2. P. Biler and G. Karch, A Neumann problem for a convection-diffusion equation on the half-line, Ann. Polon. Math. 74 (2000), 79-95. https://doi.org/10.4064/ap-74-1-79-95
  3. H. Brezis, L. A. Peletier, and D. Terman, A very singular solution of the heat equation with absorption, Arch. Rational Mech. Anal. 95 (1986), no. 3, 185-209.
  4. J. L. Diaz and J. E. Saa, Uniqueness of very singular self-similar solution of a quasilinear degenerate parabolic equation with absorption, Publ. Mat. 36 (1992), no. 1, 19-38. https://doi.org/10.5565/PUBLMAT_36192_02
  5. E. DiBenedetto, Degenerate Parabolic Equations, Springer-Verlag, New York, 1993.
  6. M. Escobedo, O. Kavian, and H. Matano, Large time behavior of solutions of a dissipative semilinear heat equation, Comm. Partial Differential Equations 20 (1995), no. 7-8, 1427-1452. https://doi.org/10.1080/03605309508821138
  7. M. Escobedo and E. Zuazua, Large time behavior for convection-diffusion equations in $R^N$, J. Funct. Anal. 100 (1991), no. 1, 119-161. https://doi.org/10.1016/0022-1236(91)90105-E
  8. M. Escobedo, J. L. Vazquez, and E. Zuazua, A diffusion-convection equation in several space dimensions, Indiana Univ. Math. J. 42 (1993), no. 4, 1413-1440. https://doi.org/10.1512/iumj.1993.42.42065
  9. L. C. Evans, Partial Differential Equations, American Mathematical Society, Providence, RI, 1998.
  10. Z. B. Fang and M. Kwak, Complete classification of shape functions of self-similar solutions, J. Math. Anal. Appl. 330 (2007), no. 2, 1447-1464. https://doi.org/10.1016/j.jmaa.2006.08.042
  11. M. Guedda, Self-similar solutions to a convection-diffusion processes, Electron. J. Qual. Theory Differ. Equ. 2000 (2000), no. 3, 17 pp.
  12. S. Kamin and L. Veron, Existence and uniqueness of the very singular solution of the porous media equation with absorption, J. Analyse Math. 51 (1988), 245-258. https://doi.org/10.1007/BF02791125
  13. M. Kwak, A semilinear heat equation with singular initial data, Proc. Roy. Soc. Edinburgh Sect. A 128 (1998), no. 4, 745-758. https://doi.org/10.1017/S0308210500021752
  14. M. Kwak, A porous media equation with absorption. II. Uniqueness of the very singular solution, J. Math. Anal. Appl. 223 (1998), no. 1, 111-125.
  15. M. Kwak, A porous media equation with absorption. I. Long time behaviour, J. Math. Anal. Appl. 223 (1998), no. 1, 96-110. https://doi.org/10.1006/jmaa.1998.5961
  16. M. Kwak and K. Yu, Asymptotic behaviour of solutions of a degenerate parabolic equation, Nonlinear Anal. 45 (2001), no. 1, Ser. A: Theory Methods, 109-121. https://doi.org/10.1016/S0362-546X(99)00333-8
  17. Ph. Laurencot and F. Simondon, Source-type solutions to porous medium equations with convection, Commun. Appl. Anal. 1 (1997), no. 4, 489-502.
  18. Ph. Laurencot and F. Simondon, Long-time behaviour for porous medium equations with convection, Proc. Roy. Soc. Edinburgh Sect. A 128 (1998), no. 2, 315-336. https://doi.org/10.1017/S0308210500012816
  19. G. Leoni, On the existence of fast-decay solutions for a quasilinear elliptic equation with a gradient term, Atti Sem. Mat. Fis. Univ. Modena 46 (1998), suppl., 827-846.
  20. L. A. Peletier and H. C. Serafini, A very singular solution and other self-similar solutions of the heat equation with convection, Nonlinear Anal. 24 (1995), no. 1, 29-49. https://doi.org/10.1016/0362-546X(95)90094-Z
  21. L. A. Peletier and D. Terman, A very singular solution of the porous media equation with absorption, J. Differential Equations 65 (1986), no. 3, 396-410. https://doi.org/10.1016/0022-0396(86)90026-4
  22. L. A. Peletier and J.Wang, A very singular solution of a quasilinear degenerate diffusion equation with absorption, Trans. Amer. Math. Soc. 307 (1988), no. 2, 813-826. https://doi.org/10.1090/S0002-9947-1988-0940229-6
  23. Z. Q. Wu, J. N. Zhao, J. X. Yin, and H. L. Li, Nonlinear Diffusion Equations, WorldScientific Publishing Co., Inc., River Edge, NJ, 2001.
  24. J. N. Zhao, The asymptotic behaviour of solutions of a quasilinear degenerate parabolic equation, J. Differential Equations 102 (1993), no. 1, 33-52. https://doi.org/10.1006/jdeq.1993.1020