×

zbMATH — the first resource for mathematics

Attractivity properties of nonnegative solutions for a class of nonlinear degenerate parabolic problems. (English) Zbl 0556.35083
The authors study the large time behavior of nonnegative solutions to an initial-boundary value problem. Using monotonicity methods they investigate attractivity properties to the associated stationary problem. Finally they apply the results to two models suggested by population dynamics.

MSC:
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
92D25 Population dynamics (general)
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35K65 Degenerate parabolic equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] H. W.Alt - S.Luckhaus,Quasi-linear elliptic-parabolic differential equations, Sonderforschungsbereich 123, Preprint n. 136, University of Heidelberg, 1981. · Zbl 0497.35049
[2] Amann, H., On the existence of positive solutions of nonlinear boundary value problems, Indiana Univ Math. J., 21, 125-146 (1971) · Zbl 0209.13002
[3] Aronson, D. G.; Crandall, M. G.; Peletier, L. A., Stabilization of solutions of a degenerate nonlinear diffusion problem, Nonl. Anal. TMA, 6, 1001-1022 (1982) · Zbl 0518.35050
[4] Aronson, D. G.; Peletier, L. A., Large time behaviour of solutions of the porous medium equation in bounded domains, J. Differential Equations, 39, 378-412 (1981) · Zbl 0475.35059
[5] Berestycky, H.; Lions, P. L.; Bardos, C.; Lasry, M.; Schatzmann, M., Some applications of the method of super and subsolutions, Bifurcation and Nonlinear Eigenvalue Problems, 16-41 (1980), Berlin-Heibelberg-New York: Springer, Berlin-Heibelberg-New York
[6] Brézis, H.; Crandall, M. G., Uniqueness of solutions of the initial-value problem for u_t-Δϕ(u)=0, J. Math. Pures Appl., 58, 153-163 (1979) · Zbl 0408.35054
[7] Crandall, M. G.; Pierre, M., Regularizing effects for u_t+Aϕ(u)=0 in L^1, Journ. Funct. Anal., 45, 2, 194-212 (1983)
[8] R.Dal Passo,Multiplicity and stability of equilibrium solutions of a one dimensional fast-diffusion problem, Boll. U.M.I., 1984 (in print). · Zbl 0557.35070
[9] Fife, P. C., The mathematics of reacting and diffusing systems, Lecture Notes in Biomathematics, Vol. 28 (1979), Berlin-Heidelberg-New York: Springer, Berlin-Heidelberg-New York · Zbl 0403.92004
[10] Gilding, B. G.; Peletier, L. A., Continuity of solutions of the porous media equations, Annali Scuola Norm. Sup. Pisa, Ser. IV, 8, 4, 659-675 (1981) · Zbl 0481.35026
[11] Gurtin, M. E.; Maccamy, R. C., On the diffusion of biological populations, Math. Biosci., 33, 35-49 (1977) · Zbl 0362.92007
[12] Lions, P. L., On the existence of positive solutions in semilinear elliptic equations, SIAM Review, 24, 4, 441-467 (1982) · Zbl 0511.35033
[13] Okubo, A., Diffusion and ecological problems: mathematical models, Biomathematics, Vol. 10 (1980), Berlin-Heidelberg-New York: Springer, Berlin-Heidelberg-New York · Zbl 0422.92025
[14] Peletier, L. A.; Amann, H.; Bazley, N.; Kirchgässner, K., The porous medium equation, Applications of Nonlinear Analysis in the Physical Sciences, 229-241 (1981), London: Pitman, London
[15] Sabinina, E. S., A class of nonlinear degenerating parabolic equations, Sov. Math. Dokl., 143, 495-498 (1962) · Zbl 0122.33503
[16] Stakgold, I.; Payne, L. E.; Stakgold, I.; Joseph, D. D.; Sattinger, D. H., Nonlinear problems in nuclear reactor analysis, Nonlinear Problems in the Physical Sciences and Biology, 298-307 (1973), Berlin-Heidelberg-New York: Springer, Berlin-Heidelberg-New York
[17] Vainberg, M. M., Variational methods for the study of nonlinear operators (1964), San Francisco: Holden Day, San Francisco · Zbl 0122.35501
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.