The asymptotic behavior of the solutions of degenerate parabolic equations

Bandle, Catherine; Pozio, M. A.; Tesei, Alberto

doi:10.1090/S0002-9947-1987-0902780-3

The asymptotic behavior of the solutions of degenerate parabolic equations
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by Catherine Bandle, M. A. Pozio and Alberto Tesei PDF

Trans. Amer. Math. Soc. 303 (1987), 487-501 Request permission

Abstract:

Existence of stationary states is established by means of the method of upper and lower solutions. The structure of the solution set is discussed and a uniqueness property for certain classes is proved by a generalized maximum principle. It is then shown that all solutions of the parabolic equation converge to a stationary state.

References

Donald Aronson, Michael G. Crandall, and L. A. Peletier, Stabilization of solutions of a degenerate nonlinear diffusion problem, Nonlinear Anal. 6 (1982), no. 10, 1001–1022. MR 678053, DOI 10.1016/0362-546X(82)90072-4
Emmanuele DiBenedetto, Continuity of weak solutions to a general porous medium equation, Indiana Univ. Math. J. 32 (1983), no. 1, 83–118. MR 684758, DOI 10.1512/iumj.1983.32.32008
J. Ildefonso Diaz and Jesús Hernández, On the existence of a free boundary for a class of reaction-diffusion systems, SIAM J. Math. Anal. 15 (1984), no. 4, 670–685. MR 747428, DOI 10.1137/0515052
Theodore Laetsch, Uniqueness for sublinear boundary value problems, J. Differential Equations 13 (1973), 13–23. MR 369899, DOI 10.1016/0022-0396(73)90028-4
Morton E. Gurtin and Richard C. MacCamy, On the diffusion of biological populations, Math. Biosci. 33 (1977), no. 1-2, 35–49. MR 682594, DOI 10.1016/0025-5564(77)90062-1
P. de Mottoni, A. Schiaffino, and A. Tesei, Attractivity properties of nonnegative solutions for a class of nonlinear degenerate parabolic problems, Ann. Mat. Pura Appl. (4) 136 (1984), 35–48. MR 765914, DOI 10.1007/BF01773375
L. A. Peletier and A. Tesei, Global bifurcation and attractivity of stationary solutions of a degenerate diffusion equation, Adv. in Appl. Math. 7 (1986), no. 4, 435–454. MR 866703, DOI 10.1016/0196-8858(86)90024-2
Maria Assunta Pozio and Alberto Tesei, Support properties of solutions for a class of degenerate parabolic problems, Comm. Partial Differential Equations 12 (1987), no. 1, 47–75. MR 869102, DOI 10.1080/03605308708820484
Toshiyuki Namba, Density-dependent dispersal and spatial distribution of a population, J. Theoret. Biol. 86 (1980), no. 2, 351–363. MR 585955, DOI 10.1016/0022-5193(80)90011-9
Paul E. Sacks, The initial and boundary value problem for a class of degenerate parabolic equations, Comm. Partial Differential Equations 8 (1983), no. 7, 693–733. MR 700733, DOI 10.1080/03605308308820283
M. Schatzman, Stationary solutions and asymptotic behavior of a quasilinear degenerate parabolic equation, Indiana Univ. Math. J. 33 (1984), no. 1, 1–29. MR 726104, DOI 10.1512/iumj.1984.33.33001
Joel Spruck, Uniqueness in a diffusion model of population biology, Comm. Partial Differential Equations 8 (1983), no. 15, 1605–1620. MR 729195, DOI 10.1080/03605308308820317