Local existence, uniqueness and regularity for a class of degenerate parabolic systems arising in biological models

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1. [1] S. Agmon, On the eigenfunctions and the eigenvalues of general elliptic boundary value problems, Comm. Pure Appl. Math., 15 (1962), pp. 119-147. Zbl0109.32701MR147774
2. [2] W. Alt, Models for mutual attractions and aggregations of motile individuals, L.N.B. 57, Proceedings of Mathematics in Biology and Medicine, Capasso-Grosso-Papaveri Fontana editors, Bari (1983). Zbl0591.92025
3. [3] D.G. Aronson - A. Tesei - H. Weinberger, A density dependent diffusion system with stable discontinuous stationary solutions, Ann. Mat. Pura Appl., to appear. Zbl0673.35054
4. [4] M. Bertsch - M.E. Gurtin - D. Hilhorst - L.A. Peletier, On interacting populations that disperse to avoid crowding: the effect of a sedentary colony, J. Math. Biology, 19 (1984), pp. 1-12. Zbl0538.92020MR737167
5. [5] M. Bertsch - M.E. Gurtin - D. Hilhorst - L.A. Peletier, On interacting populations that disperse to avoid crowding: preservation of segregation, J. Math. Biology, 23 (1985), pp. 1-13. Zbl0596.35074MR821681
6. [6] M. Bertsch - D. Hilhorst, A density dependent diffusion problem in population dynamics, SIAM J. Math. Anal., 17 (1986), pp. 863-883. Zbl0607.35052MR846394
7. [7] M. Bertsch - D.E. Gurtin - D. Hilhorst, On a degenerate diffusion equation of the form c(z)t = Φ(zx)x with applications to population dynamics, J. Diff. Eqs., 67 (1987), pp. 56-89. Zbl0624.35049
8. [8] S. Campanato - G. Stampacchia, Sulle maggiorazioni in Lp nella teoria delle equazioni ellittiche, Boll. Un. Mat. Ital., 20 (1965), pp. 393-399. Zbl0142.37604MR192169
9. [9] P. Cannarsa - B. TERRENI - V. VESPRI, Analytic semigroups generated by nonvariational elliptic systems of second order under Dirichlet boundary conditions, J. Math. Anal. Appl., 112 (1985), pp. 56-103. Zbl0593.47048MR812793
10. [10] P. Cannarsa - V. Vespri, Generation of analytic semigroups in the LP topology by elliptic operators in Rn, Israel J. Math., 61 (1988), pp. 235-255. Zbl0669.35026MR941240
11. [11] V. Capasso- S.L. Paveri Fontana, A mathematical model for the 1973 cholera epidemic in the european mediterranean region, Rev. Epidém. Santé Publ., 27 (1979), pp. 121-132. 
12. [12] V. Capasso - B. Grosso - S.L. Paveri Fontana, Proceedings of mathematics in Biology and Medicine, L.N.S. 57, Bari (1983). Zbl0559.00025
13. [13] R. Dal Passo - P. De Mottoni, On existence, uniqueness and attractivity of stationary solutions to some quasilinear parabolic systems, L.B.N. 57, Proceedings of Mathematics in Biology and Medicine, Capasso- Grosso-Paveri Fontana editors, Bari (1983). Zbl0572.35035
14. [14] A. Fasano - M. Primicerio, Nonlinear diffusion problems, L.N.S. 1224, Corso CIME, Montecatini Terme (1985). Zbl0595.00014
15. [15] E.F. Keller - L. A. SEGEL, Model of chemotaxis, J. Theor. Biol., 30 (1971), pp. 225-234. Zbl1170.92307
16. [16] J.L. Lions, Théoremes de trace et d'interpolation - I, Ann. Scuola Norm. Sup. Pisa, 13 (1959), pp. 389-403. Zbl0097.09502MR119092
17. [17] J.L. Lions - J. PEETRE, Sur une class d'espaces d'interpolation, Publ. I.H.E.S., Paris (1964), pp. 5-68. Zbl0148.11403MR165343
18. [18] A. Lunardi, On the local dynamical system associated to a fully nonlinear abstract parabolic equation, in Nonlinear Analysis and Applications, V. Laksmikantham editor (1987), pp. 319-326. Zbl0646.47045MR912310
19. [19] H. Matano - M. Mimura, Pattern formation in competition diffusion systems in nonconvex domains, Publ. Res. Inst. Math. Sci. Kyoto Univ., 19 (1983), pp. 1049-1079. Zbl0548.35063MR723460
20. [20] M. Mimura, Stationary pattern of some density dependent diffusion systems with competitive dynamics, Hiroshima Math. J., 11 (1981), pp. 621-635. Zbl0483.35045MR635042
21. [21] M.A. Pozio - A. TESEI, Degenerate parabolic problems in populaction dynamics, Japan J. Appl. Math., 2 (1985), pp. 351-380. MR839335
22. [22] M.A. Pozio - A. TESEI, Global existence of solutions for a strongly coupled quasilinear parabolic system, Nonlinear Analysis, 14 (1990), pp. 657-690. Zbl0716.35034MR1049787
23. [23] N. Shigesada - K. Kawasaki - E. Teramoto, Spatial segregation of interacting species, J. Theor. Biology, 79 (1979), pp. 83-99. MR540951
24. [24] E. Sinestrari, On the abstract Cauchy problem of parabolic type in spaces of continuous functions, J. Math. Anal. Appl., 107 (1985), pp. 16-66. Zbl0589.47042MR786012
25. [25] V. Vespri, Analytic semigroups, degenerate elliptic operators and applications to nonlinear Cauchy problems, Ann. Mat. Pura Appl., 155 (1989), pp. 353-388. Zbl0709.35065MR1042844
26. [26] W. Von Wahl, Gebrochene potenzen eines elliptischen operator und parabolische differentialgleichungen in raumen Hölderstetiger funktionen, Nachr. Akad. Wiss. Gottingen Math. Phis. Kl., 11 (1972), pp. 231-258. Zbl0251.35052MR313636