A degenerate nonlinear diffusion system with boundary feedbackFootnote^δ

^δWork performed under the auspices of the GNAFA-CNR (L.M.) and GNFM-CNR (V.C.) with the financial support of the program "Control of Infectious Diseases" (CNR-Italy) and M.P.I.-Italy.

References

• Amann , H. 1976 . “ Nonlinear elliptic equations with nonlinear boundary conditions ” . In New Developments in Differential Equations , Edited by: Eckaus , W. Amsterdam : North-Holland Publishing Company .
   Google Scholar
• Aronson , D.G. , Crandall , M.G. and Peletier , L.A. 1982 . Stabilization of solutions of a degenerate nonlinear diffusion problem . Nonlinear Analysis T.M.A. , 6 : 1001 – 1022 .
   Web of Science ®Google Scholar
• Aronson , D.G. and Peletier , L.A. 1981 . Large time behaviour of solutions of the porous medium equation in a bounded domain . J. Diff. Equations , 39 : 378 – 412 .
   Web of Science ®Google Scholar
• Bertsch , M. 1983 . Nonlinear Diffusion Problems , The Large Time Behaviour .
   Google Scholar
• Bertsch , M. , Nanbu , T. and Peletier , L.A. 1982 . Decay of solutions of a degenerate nonlinear diffusion equations . Nonlinear Analysis T.M.A. , 6 : 539 – 554 .
   Google Scholar
• Capasso , V. and Kunish , K. 1985 . A reaction-diffusion system modelling man-environment epidemics . Annals of Differential Equations , 1 : 1 – 12 .
   Google Scholar
• Capasso , V. and Maddalena , L. 1981 . Convergence to equilibrium states for a reaction-diffusion system modelling the spatial spread of a class of bacterial and viral diseases . J. Math. Biology , 13 : 173 – 184 .
   PubMed Web of Science ®Google Scholar
• Capasso , V. and Paveri-Fontana , S.L. . A mathematical model for the 1973 cholera epidemic in the European Mediterranean region . Rev, Epidem. Santè Publ . Ibidem. Vol. 28 , pp. 390 – 390 . Errata .
   Google Scholar
• Di Benedetto , E. 1983 . Continuity of weak solutions to a general porous medium equation . Indiana Univ. Math. J. , 32 : 83 – 118 .
   Web of Science ®Google Scholar
• Friedman , A. 1969 . Partial Differential Equations , New York : Rinehart and Winston Inc. . Holt
   Google Scholar
• Gilding , B.H. and Peletier , L.A. 1981 . Continuity of solutions of the porous media equation . Ann. Scuola Norm. Sup. Pisa , VIII : 659 – 675 .
   Google Scholar
• Gurney , W.S.C. and Nisbet , R.M. 1975 . The regulation of inhomoge-neous populations . J. Theor. Biol. , 52 : 441 – 457 .
   PubMed Web of Science ®Google Scholar
• Gurtin , E. and Mac Camy , R.C. 1977 . On the diffusion of biological populations . Math. Biol. , 33 : 35 – 49 .
   Google Scholar
• Ladyzhenskaja , O.A. , Solonnikov , V.A. and Ural'ceva , N.N. 1983 . Linear and Quasilinear Equations of Parabolic Type, Transla-tions of Mathematical Monographs 23 , Vol. 23 , A.M.S. Providence RI .
   Google Scholar
• Maddalena , L. 1984 . Existence of global solutions for reaction--diffusion systems with density dependent diffusion . Nonlinear Analysis T.M.A. , 8 : 1383 – 1394 .
   Web of Science ®Google Scholar
• Maddalena , L. 1986 . Existence.uniqueness and qualitative properties of the solution of a degenerate nonlinear parabolic system , To appear in J. of Math. Analysis and Appl .
   Google Scholar
• Mc Nabb , A. 1961 . Comparison and existence theorems for multicom-ponent diffusion systems . J. Math. Anal. Appl. , 8 : 133 – 144 .
   Google Scholar
• Okubo , A. 1980 . “ Diffusion and Ecology Problems ” . In Mathematical Models , Heidelberg : Springer-Verlag .
   Google Scholar
• Oleinik , Kalashnikov , A.S. and Yui-Lin Chzou . 1958 . The Cauchy problem and boundary value problems for equations of the type of non stationary filtration . Izv. Akad. Nauk. S.S.S.R., Soc. Mat. , 22 : 667 – 704 .
   Google Scholar
• Puel , J.P. 1976 . Existence, comportement a l'infini et stabilité dans certaine problémss quasilineaires elliptiques et paraboliques d'ordre 2 . Ann. Sc. Norm. Sup. , 3 : 89 – 119 . Pisa
   Google Scholar
• Sabinina , E.S. 1961 . On the Cauchy problem for the equation of nonstationary gas filtration in several space variables . Dokl. Akad. Nauk S.S.S.R. , 136 : 1034 – 1037 .
   Google Scholar
• Sacks , P.E. 1983 . Continuity of solutions of a singular parabolic equation . Nonlinear Analysis, T.M.A. , 7 : 387 – 409 .
   Web of Science ®Google Scholar
• Sattinger , D.H. 1972 . Monotone methods in nonlinear elliptic and parabolic boundary value problems . Indiana Univ. Math. J. , 21 : 979 – 1000 .
   Web of Science ®Google Scholar
• Vainberg , M.M. 1973 . Variational Method and Method of Monotone Operators in the Theory of Nonlinear Equations , N.Y : Halsted Press, John Wiley and Sons Inc. .
   Google Scholar