The number of peaks of nonnegative solutions to some nonlinear degenerate parabolic equations

Journal of Mathematical Analysis and Applications

Volumn 203, Issue 1, 1996, Pages 78-103

  Sakaguchi, Shigeru ^a  

^a EHIME UNIVERSITY   (Japan)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0030268367     PISSN: 0022247X     EISSN: None     Source Type: Journal    
DOI: 10.1006/jmaa.1996.0368     Document Type: Article

References (53)

1. 1. S. Angenent, The zero set of a solution of a parabolic equation, J. Reine Angew. Math. 390 (1988), 79-96.
2. 2. D. G. Aronson, The porous medium equation, in "Nonlinear Diffusion Problems" (A. Fasano and M. Primicerio, Eds.), pp. 1-46, Lecture Notes in Math., Vol. 1224, Springer-Verlag, New York/Berlin, 1986.
3. N, C. R. Acad. Sci. Paris Sér. A 288 (1979), 103-105.
4. 4. D. G. Aronson and L. A. Caffarelli, The initial trace of a solution of the porous medium equation, Trans. Amer. Math. Soc. 280 (1983), 351-366.
5. 5. D. G. Aronson and L. A. Peletier, Large time behaviour of solutions of the porous medium equation in bounded domains, J. Differential Equations 39 (1981), 378-412.
6. ∞-regularity and concavity for flows in one-dimensional porous media, Arch. Rational Mech. Anal. 99 (1987), 329-348.
7. 7. M. Bardi, Asymptotic spherical symmetry of the free boundary in degenerate diffusion equations, Ann. Mat. Pura Appl. 148 (1987), 117-130.
8. t - Δφ(u) = 0, Indiana Univ. Math. J. 30 (1981), 161-177.
9. N under optimal conditions on initial values, Indiana Univ. Math. J. 33 (1984), 51-87.
10. 10. M. Bertsch and D. Hilhorst, The interface between regions where u < 0 and u > 0 in the porous medium equation, Appl. Anal. 41 (1991), 111-130.
11. t - Δφ(u) = 0, J. Math. Pures Appl. 58 (1979), 153-163.
12. 12. L. A. Caffarelli, J. L. Vazquez, and N. I. Wolanski, Lipschitz continuity of solutions and interfaces of the N-dimensional porous medium equation, Indiana Univ. Math. J. 36 (1987), 373-401.
13. 13. L. A. Caffarelli and A. Friedman, Regularity of the free boundary of a gas flow in an n-dimensional porous medium, Indiana Univ. Math. J. 29 (1980), 361-391.
14. 1, α regularity of the free boundary for the N-dimensional porous media equation, Comm. Pure Appl. Math. 43 (1990), 885-902.
15. 15. I. Chavel and L. Karp, Movement of hot spots in Riemannian manifolds, J. Analyse Math. 55 (1990), 271-286.
16. 16. Y. Z. Chen and E. DiBenedetto, On the local behavior of solutions of singular parabolic equations, Arch. Rational Mech. Anal. 103 (1988), 319-345.
17. 17. B. E. J. Dahlberg and C. E. Kenig, Non-negative solutions of the porous medium equation, Comm. Partial Differential Equations 9 (1984), 409-437.
18. 18. B. E. J. Dahlberg and C. E. Kenig, Non-negative solutions to fast diffusions, Rev. Mat. Iberoamericana 4 (1988), 11-29.
19. 19. J. I. Diaz and M. A. Herrero, Estimates on the support of the solutions of some nonlinear elliptic and parabolic problems, Proc. Roy. Soc. Edinburgh Sect. A 89 (1981), 249-258.
20. 20. E. DiBenedetto, Continuity of weak solutions to a general porous medium equation, Indiana Univ. Math. J. 32 (1983), 83-118.
21. 21. E. DiBenedetto, On the local behaviour of solutions of degenerate parabolic equations with measurable coefficients, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 13 (1986), 487-535.
22. 22. E. DiBenedetto and A. Friedman, Hölder estimates for nonlinear degenerate parabolic systems, J. Reine Angew. Math. 357 (1985), 1-22.
23. 23. E. DiBenedetto and A. Friedman, Addendum to "Hölder estimates for nonlinear degenerate parabolic systems," J. Reine Angew. Math. 363 (1985), 217-220.
24. 24. E. DiBenedetto and M. A. Herrero, On the Cauchy problem and initial traces of a degenerate parabolic equation, Trans. Amer. Math. Soc. 314 (1989), 187-224.
25. 25. J. R. Esteban and J. L. Vazquez, On the equation of turbulent filtration in one-dimensional porous media, Nonlinear Anal. 10 (1986), 1303-1325.
26. 26. J. R. Esteban and J. L. Vazquez, Homogeneous diffusion in ℝ with power-like nonlinear diffusivity, Arch. Rational Mech. Anal. 103 (1988), 39-80.
27. 27. A. Friedman, "Partial Differential Equations of Parabolic Type," Prentice-Hall, Englewood Cliffs, NJ, 1964.
28. 28. A. Friedman and S. Kamin, The asymptotic behavior of gas in an n-dimensional porous medium, Trans. Amer. Math. Soc. 262 (1980), 551-563.
29. 29. B. Gidas, W. M. Ni, and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979), 209-243.
30. 30. D. B. Henry, Some infinite-dimensional Morse-Smale systems defined by parabolic partial differential equations, J. Differential Equations 59 (1985), 165-205.
31. m) when 0 < m < 1, Trans. Amer. Math. Soc. 291 (1985), 145-158.
32. 32. M. A. Herrero and J. L. Vazquez, On the propagation properties of a nonlinear degenerate parabolic equation, Comm. Partial Differential Equations 7 (1982), 1381-1402.
33. N, J. Math. Anal. Appl. 182 (1994), 810-835.
34. 34. A. S. Kalashnikov, The occurrence of singularities in solutions of the non-steady seepage equation, Zh. Vychisl. Mat. i Mat. Fiz. 7 (1967), 440-444; U.S.S.R. Comput. Math. and Math. Phys. 7 (1967), 269-275. Translation.
35. 34. A. S. Kalashnikov, The occurrence of singularities in solutions of the non-steady seepage equation, Zh. Vychisl. Mat. i Mat. Fiz. 7 (1967), 440-444; U.S.S.R. Comput. Math. and Math. Phys. 7 (1967), 269-275. Translation.
36. 35. A. S. Kalashnikov, Some problems of the qualitative theory of non-linear degenerate second-order parabolic equations, Russian Math. Surveys 42 (1987), 169-222.
37. 36. S. Kamin and J. L. Vazquez, Fundamental solutions and asymptotic behaviour for the p-Laplacian equation, Rev. Mat. Iberoamericana 4 (1988), 339-354.
38. 37. S. Kamin and J. L. Vazquez, Asymptotic behaviour of solutions of the porous medium equation with changing sign, SIAM J. Math. Anal. 22 (1991), 34-45.
39. 38. M. H. A. Klaver, The Stefan problem: Continuity of the interfaces for solutions with finite lapnumber, Proc. Roy. Soc. Edinburgh Sect. A 123 (1993), 1-26.
40. 39. B. F. Knerr, The porous medium equation in one dimension, Trans. Amer. Math. Soc. 234 (1977), 381-415.
41. 40. Y. C. Kwong, Interior and boundary regularity of solutions to a plasma type equation, Proc. Amer. Math. Soc. 104 (1988), 472-478.
42. 41. O. A. Ladyženskaja, V. A. Solonnikov, and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type," Transl. Math. Monographs, Vol. 23, Amer. Math. Soc., Providence, 1968.
43. 42. G. M. Lieberman, Initial regularity for solutions of degenerate parabolic equations, Nonlinear Anal. 14 (1990), 525-536.
44. 43. H. Matano, Nonincrease of the lap-number of a solution for a one-dimensional semilinear parabolic equation, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 29 (1982), 401-441.
45. 44. H. Matano, Asymptotic behaviour of the free boundaries arising in one phase Stefan problems in multi-dimensional spaces, in "Lecture Notes Numer. Appl. Anal.," Vol. 5, pp. 133-151, Kinokuniya, Tokyo, 1982.
46. 45. W. M. Ni and P. Sacks, The number of peaks of positive solutions of semilinear parabolic equations, SIAM J. Math. Anal. 16 (1985), 460-471.
47. 46. L. A. Peletier and Z. Junning, Large time behaviour of solutions of the porous media equation with absorption: The fast diffusion case, Nonlinear Anal. 17 (1991), 991-1009.
48. 47. M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations," Prentice-Hall, Englewood Cliffs, NJ, 1967.
49. 48. E. S. Sabinina, A class of nonlinear degenerating parabolic equations, Soviet Math. Dokl. 143 (1962), 495-498.
50. 49. P. Sacks, Continuity of solutions of a singular parabolic equation, Nonlinear Anal. 7 (1983), 387-409.
51. 50. J. L. Vazquez, Asymptotic behaviour and propagation properties of the one-dimensional flow of gas in a porous medium, Trans. Amer. Math. Soc. 277 (1983), 507-527.
52. 51. J. L. Vazquez, The interfaces of one-dimensional flows in porous media, Trans. Amer. Math. Soc. 285 (1984), 717-737.
53. α-regularity of the gradient of solutions of degenerate parabolic systems, Ann. Mat. Pura Appl. 145 (1986), 385-405.