Nonlinear systems of partial differential equations: Applications to life and physical sciences

Nonlinear Systems of Partial Differential Equations: Applications to Life and Physical Sciences

Volumn , Issue , 2009, Pages 1-532

  Leung, Anthony W ^a  

^a UNIVERSITY OF CINCINNATI   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

BOUNDARY CONDITIONS; MAXWELL EQUATIONS; NONLINEAR EQUATIONS; NUCLEAR REACTORS; PLASMA DISPLAY DEVICES;

DIFFUSION-REACTION EQUATION; DIRICHLET BOUNDARY CONDITION; LARGE COMPLEX SYSTEMS; NON-LINEAR METHODS; RESOURCE MANAGEMENT; STABILITY THEOREMS; SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONS; UPPER-LOWER SOLUTION;

NAVIER STOKES EQUATIONS;

EID: 84967588147     PISSN: None     EISSN: None     Source Type: Book    
DOI: 10.1142/7353     Document Type: Book

References (244)

1. S. Agmon, A. Douglis, and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions I, Comm. Pure Appl. Math. 12 (1959), 623-727.
2. N. Alikakos, An application of the invariance principle to reaction-diffusion equations, J. Diff. Eqs. 33 (1979), 201-225.
3. H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev. 18 (1976), 620-709.
4. P. Angelstam, Predation on ground-nesting birds’ nests in relation to predator densities and habitat edge, Oikos, 47 (1986), 365-373.
5. S. Anita, Analysis and Control of Age-Dependent Population Dynamics, Mathematical Modelling: Theory and Applications, Vol. 11, Kluwer Academic Publishers, Boston, 2000.
6. D. Aronson, M. Crandall and L. Peletier, Stabilization of solutions of a degenerate nonlinear diffusion problem, Nonlinear Anal. T.M.A. 6 (1982), 1001-1022.
7. D. Aronson and H. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve propagation, Lecture Notes in Mathematics, 446, 5-49, Springer, New York (1975).
8. H. Berestycki and P. Lions, Some applications of the method of sub- and supersolutions, Bifurcation and Nonlinear Eigenvalue Problems, p. 16-41, Lecture Notes in Mathematics 782, Springer-Verlag, Berlin 1980.
9. N. Bhatia and G. Szego, Stability Theory of Dynamical Systems, Springer, Berlin, 1970.
10. R. Bierregaard, T. Lovejoy, T. Kapos, A dos Santos and R. Hutchings, The biological dynamics of tropical rain forest fragments, Biosci. 42 (1992), 859-866.
11. J. Billoti and J. P. LaSalle. Periodic dissipative processes, Bull. Amer. Math. Soc., 6 (1971), 1082-1089.
12. J. Blat and K. Brown, Bifurcation of steady-state solutions in predator- prey and competition systems, Proc. Roy. Soc. Edinburgh, Sect. A 97 (1984), 21-34.
13. J. Blat and K. Brown, A reaction-diffusion system modeling the spread of bacterial infections, Math. Methods Appl. Sci. 8 (1986), 234-246.
14. F. Brauer and J. Nohel, Qualitative Theory of Ordinary Differential Equations, W.A. Benjamin, N.Y., 1969.
15. A. Boumenir and V. Nguyen, Perron theorem in monotone iteration method for traveling waves in delayed reaction-diffusion equations, J. Diff. Eqs. 244 (2008), 1551-1570.
16. K. Brown, Spatially inhomogeneous steady state solutions for systems of equations describing interacting populations, J. Math. Anal. Appl. 95 (1983), 251-264.
17. G. Butler, H. Freedman and P. Waltman, Uniformly persistent dynamical systems, Proc. Amer. Math. Soc. 96 (1986), 425-430.
18. A. Canada, P. Drabek and A. Fonda, Handbook of Differential Equations: Ordinary Differential Equations, Vol. 2, Elsevier Science and Technology, New York, 2004.
19. R. Cantrell and C. Cosner, On the steady state problem for the Volterra- Lotka competition model with diffusion, Houston J. Math., 13 (1987), 337-352.
20. R. Cantrell and C. Cosner, Practical persistence in ecological models via comparison methods, Proc. R. Soc. Edinburgh, 126A (1996), 247-272.
21. R. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, Wiley Series in Mathematical and Computational Biology, Wiley, West Sussex, 2003.
22. R. Cantrell, C. Cosner and V. Hutson, Permanence in ecological systems with spatial heterogeneity, Proc. R. Soc. Edinburgh, 123A (1993), 533-559.
23. R. Cantrell, C. Cosner and V. Hutson, Permanence in some diffusive Lotka- Volterra models for three interacting species, Dynamic Sys. and Appl. 2 (1993), 505-530.
24. V. Capasso and L. Maddalena, A non-linear diffusion system modelling the spread of oro-faecal diseases, in “Nonlinear Phenomena in Mathematical Sciences” (V. Laksmikantham, Ed.), Academic Press, New York, 1981.
25. V. Capasso and L. Maddalena, Convergence to equilibrium states for a reaction-diffusion system modelling the spread of a class of bacterial and viral diseases, J. Math. Biology 13 (1981), 173-184.
26. A. Castro and A. Lazer, Results on periodic solutions of parabolic equations suggested by elliptic theory, Boll. Un. Mat. Ital. B(I) 6 (1982), 1089-1104.
27. S. Chen and S.T. Yau, Ed., Geometry and Nonlinear Partial Differential Equations, AMS/IP Studies in Advanced Mathematics Series, 2002.
28. M. Chipot, M. Fila and P. Quittner, Stationary solutions, blowup and convergence to stationary solutions for semilinear parabolic equations with nonlinear boundary conditions, Acta Math. Univ. Comenianae, LX (1991), 35-105.
29. S. Chow and J. Hale, Methods of Bifurcation Theory, Springer-Verlag, N.Y., 1982.
30. G. Christensen, S. Soliman and R. Nieva, Optimal Control of Distributed Nuclear Reactor, Plenum, New York, 1990.
31. C. Clark, Mathematical Bioeconomics: The Optimal Management of Renewable Resources, Wiley, New York, 1976.
32. C. Conley and R. Gardner, An application of generalized Morse index to traveling wave solutions of a competitive reaction diffusion model, Indiana Univ. Math. J. 33 (1984), 319-345.
33. C. Cosner and A. Lazer, Stable coexistence states in the Volterra-Lotka competition model with diffusion, SIAM J. Appl., 44 (1984), 1112-1132.
34. M. Crandall and P. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal. 8 (1971), 321-340.
35. M. Crandall and P. Rabinowitz, Bifurcation, perturbation of simple eigen-values and linearized stability, Arch. Rat. Mech. Anal. 52 (1973), 161-181.
36. S. Cui, Analysis of a mathematical model for the growth of tumors under the action of external inhibitors, J. Math. Biol. 44 (2002), 395-426.
37. S. Cui and A. Friedman, Analysis of a mathematical model of the growth of necrotic tumors, J. Math. Anal. Appl. 255 (2001), 636-677.
38. E. N. Dancer, On the indices of fixed points of mappings in cones and applications, J. Math. Anal. Appl. 91 (1983), 131-151.
39. E. N. Dancer, Counterexamples to some conjectures on the number of solutions of nonlinear equations, Math. Ann. 272 (1985), 421-440.
40. E. N. Dancer, Multiple fixed points of positive maps, J. Reine Ang. Math. 371 (1986), 46-66.
41. E. N. Dancer, On the existence and uniqueness of positive solutions for competing species models with diffusion, Trans. Amer. Math. Soc. 36 (1991), 829-859.
42. E. Dancer and Y. Du, Competing species equations with diffusion, large interactions and jumping nonlinearities, J. Diff. Eqs. 114 (1994), 434-475.
43. E. Dancer and Z. Guo, Uniqueness and Stability for solutions of competing species equations with large interactions, Comm. Appl. Nonlin. Anal. 1 (1994), 19-45.
44. P. Darlington, Zoogeography: The Geographical Distribution of Animals, Wiley, New York, 1957.
45. C. DeLisi, Antigen Antibody Interactions, Lecture Notes in Biomath., Vol. 8, Springer-Verlag, New York, 1976.
46. M. Delgado, J. Lopez-Gomez and A. Suarez, On the symbyotiic Lotka- Volterra model with diffusion and transport effects, J. Diff. Eqs. 160 (2000) 175-262.
47. P. DeMottoni, A. Schiaffino and A. Tesei, Attractive properties of nonnegative solutions of a class of nonlinear degenerate parabolic problems, Ann. Mat. Pura. Appl. 36 (1984), 35-48.
48. J. Diaz and J. Hernandez, On the existence of a free boundary for a class of reaction-diffusion systems, SIAM J. Math. Anal. 15 (1984), 670-685.
49. O. Diekmann, R. Durrett, K. Hadeler, R. Maini and H. Smith, Mathematics Inspired by Biology, Springer-Verlag, Berlin, 2000.
50. K. Diemling, Nonlinear Functional Analysis, Springer-Verlag, New York, 1985.
51. X. Ding and T. P. Liu, Ed., Nonlinear Evolutionary Partial Differential Equations, AMS/IP Studies in Advanced Mathematics Series, 1997.
52. D. Doak, Source-sink models and the problem of habitat degradation: General models and applications to Yellowstone Grizzly, Conservation Biol. 9 (1995), 1370-1379.
53. J. Duderstadt and L. Hamilton, Nuclear Reactor Analysis, John Wiley, New York, 1976.
54. J. Dugundji, An extension of Tietze’s theorem, Pacific J. Math. 1 (1951), 353-367.
55. J. Dugundji, Topology, Allyn and Bacon, Boston, 1966.
56. S. Dunbar, Traveling wave solutions of diffusive Lotka-Volterra equations, J. Math. Biol. 17 (1983), 11-32.
57. S. Dunbar, Traveling wave solutions of diffusive Lotka-Volterra equations: A heteroclinic connection in R4, Trans. Am. Math. Soc., 286 (1984), 557-594.
58. L. Evans, Partial Differential Equations, Graduate Studies in Math. 19, Amer. Math. Soc., R. I., 1998.
59. P. Fife, Solutions of parabolic boundary problems existing for all time, Arch. Rational Mech. Anal. 16 (1964), 155-186.
60. P. Fife, Mathematical Aspects of Reacting and Diffusing Systems, Lecture Notes in Biomathematics 28, Springer, Berlin, 1979.
61. M. Fila and P. Quitter, The blow up rate for heat equation with nonlinear boundary condition, Math. Meth. Appl. Sci. 14 (1991), 197-205.
62. J. Filo, Diffusivity verus absorption through the boundary, J. Diff. Eqs. 99 (1992), 281-305.
63. W. Fleming, A selection-migration model in population genetics, J. Math. Biol. 2 (1975).
64. A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, 1964.
65. A. Friedman and F. Reitich, Analysis of a mathematical model for the growth of tumors, J. Math. Biol. 38 (1999), 262-284.
66. L. Friesen, P. Eagles and R. MacKay, Effects of residential development on forest-dwelling neotropical migrant songbirds, Conservation Biol. 9 (1995), 1408-1411.
67. Y. Furusho, Existence of positive entire solutions for weakly coupled semi- linear elliptic systems, Proc. Roy. Soc. Edinburgh, 120A (1992), 79-91.
68. G. Galdi, An Introduction to Mathematical Theory of Navier-Stokes Equa-tions, Springer Tracts in Natural Philosophy, Vols 38, 39, Springer, NY, 1994.
69. R. Gardner, Existence and stability of traveling wave solutions of competition models: A degree theoretic approach, J. Diff. Eqs. 44 (1982), 56-79.
70. R. Gardner, Review on traveling wave solutions of parabolic systems by A.I. Volpert, V.A. Volpert and V.A. Volpert, Bull. Amer. Math. Soc. 32 (1995), 446-452.
71. B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Diff. Eqs. 6 (1981), 883-901.
72. D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Second Ed., Springer-Verlag, New York, 1983.
73. S. Glasstone and A. Sesonske, Nuclear Reactor Engineering, Van Nostrand, New York, 1981.
74. B. Goh, Sector stability of complex ecosystem model, Math. Biosci. 40 (1978), 157-166.
75. J. Goldstein, Semigroups of Linear Operators and Applications, Oxford Mathematical Monographs, Oxford University Press, New York, 1985.
76. A. Granas, Points fixes pour less applications compact, University of Montreal Press, Montreal, 1980.
77. M. Gurtin, Some mathematical models for population dynamics that lead to segregation, Quart. J. Appl. Math. 32 (1974), 1-9.
78. G. Hagelaar, Modeling of microdischarges for display technology, Proefschrift, Technische Universiteit Eindhoven, The Netherlands, 2000.
79. G. Hagelaar, M. Klein, R. Snijkers and G. Kroesen, Energy loss mechanisms in the microdischarges in plasma display panels, J. Appl. Phys 89 (2001), 2033-2039.
80. J. Hale, Ordinary Differential Equations, Wiley Interscience, NY, 1969.
81. J. Hale, Large diffusivity and asymptotic behaviour in parabolic systems, J. Math. Anal. Appl. 118 (1986), 455-466.
82. J. Hale and P. Waltman, Persistence in infinite-dimensional systems, SIAM J. Math. Anal. 20 (1989), 388-395.
83. F. He, A. Leung and S. Stojanovic, Periodic optimal control for parabolic Volterra-Lotka type equations, Math. Methods in Appl. Sci. 18 (1995), 127-146.
84. F. He, A. Leung and S. Stojanovic, Periodic optimal control for competing parabolic Volterra-Lotka type systems, J. of Computational and Appl. Math. 52 (1994), 199-217.
85. D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math. 840, Springer-Verlag, N.Y., 1981.
86. P. Hess, Periodic-Parabolic Boundary Value Problems and Positivity, Pitman R.N.M., Longman, Harlow, 1991.
87. M. Hirsch, Stability and convergence in strongly monotone dynamical systems, J. Reine Ang. Math. 383 (1988), 1-58.
88. J. Hofbauer and K. Sigmund, Dynamical Systems and the Theory of Evolution, Cambridge Univ. Press, Cambridge, 1988.
89. J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge Univ. Press, Cambridge, 1998.
90. F. Hoppensteadt, Asymptotic stability in singular perturbation problems, J. Diff. Eqs. 4 (1968), 350-358.
91. X. Hou and A. Leung, Traveling wave solutions for a competitive reaction- diffusion system and their asymptotics, Nonlin. Anal: Real World Appl. 9 (2008), 2196-2213.
92. B. Hu and H. Yin, The profile near blowup time for solutions of heat equation with nonlinear boundary conditions, IMA preprint series no. 1115, 1993.
93. V. Hutson, A theorem on average Liapunov functions, Monatsh. Math. 98 (1984), 267-275.
94. V. Hutson. J. Pym and M. Cloud, Applications of Functional Analysis and Operator Theory, Vol. 200, 2nd Edition, Mathematics in Science and Engineering, Elsevier Science, N.Y., 2005.
95. V. Hutson and K. Schmitt, Permanence and the dynamics of biological systems, Math. Biosci. 111 (1992), 1-71.
96. T. Jackson, Vascular tumor growth and treatment: Consequences of polyclonality, competition and dynamic vascular support, J. Math. Biol. 44 (2002), 201-226.
97. J. Jorne, The diffusive Lotka-Volterra oscillating system, J. Theor. Biol. 65 (1977), 133-139.
98. Y. Kanon, Stability of singularly perturbed solutions to nonlinear diffusion system arising in population dynamics, Hiroshima Math. J. 23 (1993), 509-536.
99. Y. Kanon, Existence and instability of Neumann layer solutions for a 3-component Lotka-Volterra model with diffusion, J. Math. Anal. Appl. 243 (2000), 357-372.
100. Y. Kanon and M. Mimura, Singular perturbation approach to a 3-component reaction-diffusion system arising in population dynamics, SIAM J. Math. Anal. 29 (1998), 1519-1536.
101. Y. Kanon and E. Yanagida, Existence of non-constant stable equilibria in competition diffusion equations, Hiroshima Math. J. 23 (1993), 193-221.
102. W. Kastenberg, Stability criterion for space-dependent nuclear reactor- systems with variable temperature feedback, Nuc. Sci. Eng. 37 (1969), 19-29.
103. T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, New York, 1966.
104. E. Kerner, Further considerations on the statistical mechanics of biological association, Bull. Math. Biophys. 21 (1959), 217-255.
105. D. Kirschner, S. Lenhart and S. Serbin, Optimal control of the chemother-apy of HIV, J. Math. Biology 35 (1997), 775-792.
106. K. Kishimoto and H. Weinberger, The spatial homogeneity of stable equi-libria of some reaction-diffusion systems on convex domains, J. Diff. Eqs. 58 (1985), 15-21.
107. A. Kolmogorov, A. Petrovskii and N. Piskunov, A study of the equation of diffusion with increase in the quantity of matter, and its application to a biological problem, Bjul. Moskovskovo Gov. Univ. 17 (1937), 1-72.
108. P. Korman and A. Leung, On the existence and uniqueness and positive steady-states in the Volterra-Lotka ecological models with diffusion, Appl. Anal. 26 (1987), 145-160.
109. M. Krasnosel’skii, Positive Solutions of Operator Equations, P. Noordhoff, Groningen, The Netherlands, 1964.
110. M. Krasnoselski and P. Zabrieko, Geometrical Methods of Nonlinear Anal-ysis, Springer-Verlag, Berlin, 1984.
111. M. Krein and M. Rutman, Linear operators leaving invariant in a cone in a Banach space, Amer. Math. Soc. Transl., 10 (1962), 199-325.
112. J. Kuby, Immunology, third ed., Freeman, New York, 1997.
113. G. Ladde, V. Lakshmikanthm and A. Vatsala, Monotone Iteration Tech-niques for Nonlinear Differential Equations, Pitman, 1985.
114. O. Ladyzhenskaya, V. Solonnikov and N. Ural’ceva, Linear and Quasilinear Equations of Parabolic Type, Amer. Math. Soc. Transl. 23, 1968.
115. I. Lasiecka and R. Triggiani, Control Problems for Systems described by Partial Differential Equations and Applications, Lecture Notes in Control and Information Sciences, No. 97, Springer-Verlag, New York, 1988.
116. A. Lazer, Some remarks on periodic solutions of parabolic equations, In Dynamical Systems II, eds A. Bednarek and L Cesari, 227-246, Academic Press, New York, 1982.
117. S. Lenhart and M. Bhat, Application of distributed parameter control model in wildlife damage management, Math. Models and Methods in Appl. Sciences 2 (1992), 423-439.
118. S. Lenhart, M. Liang and V. Protopopescu, Optimal control of boundary habitat hostility for interacting species, Math. Meth. Appl. Sci. 22 (1999), 1061-1077.
119. S. Lenhart and V. Protopopescu, Optimal control for parabolic systems with competitive interactions, Math. Meth. Appl. Sci. 17 (1994), 509-524.
120. S. Lenhart and D. Wilson, Optimal control of a heat transfer problem with convective boundary condition, J. Optim. Theory Appl. 79 (1993), 581-587.
121. S. Lenhart and J. Workman, Optimal Control Applied to Biological Models, Chapman and Hall/CRC Mathematical and Computational Biology, 2007.
122. A. Leung, Equilibria and stabilities for competing-species reaction- diffusion equations with Dirichlet boundary data, J. Math. Anal. 73 (1980), 204-218.
123. A. Leung, Stabilities for equilibria of competing-species reaction-diffusion equations with homogeneous Dirichlet condition, Funk. Ekv. (Ser. Interna.) 24 (1981), 201-210.
124. A. Leung, Monotone schemes for semilinear elliptic systems related to ecology, Math. Meth. Appl. Sci. 4 (1982), 272-285.
125. A. Leung, Nonlinear density-dependent diffusion for competing species interactions: Large-time asymptotic behaviour, Proc. Edinburgh Math. Soc. 27 (1984), 131-144.
126. A. Leung, Systems of Nonlinear Partial Differential Equations, Applications to Biology and Engineering, Kluwer Academic Publishers, Dordrecht/Boston, Springer-Verlag, N.Y., 1989.
127. A. Leung, Optimal harvesting-coefficient control of steady-state prey- predator diffusive Volterra-Lotka systems, Appl. Math. Optimization 31 (1995), 219-241.
128. A. Leung, Bifurcating positive stable steady-states for a system of wave equations, Diff. Int. Eqs. 16 (2003), 453-471.
129. A. Leung, Asymptotically stable invariant manifold for coupled nonlinear parabolic-hyperbolic partial differential equations, J. Diff. Eqs. 187 (2003), 184-200.
130. A. Leung, Positive solutions for large elliptic systems of interacting species groups by cone index methods, J. Math. Anal. Appl. 291 (2004), 302-321.
131. A. Leung, Stable invariant manifolds for coupled Navier-Stokes and second-order wave systems, Asymptotic Analysis 43 (2005), 339-357.
132. A. Leung and G. S. Chen, Positive solutions of temperature-dependent two group neutron flux equations: Equilibrium and stabilities, SIAM J. Math. Anal. 15 (1984), 131-144.
133. A. Leung and G. S. Chen, Elliptic and parabolic systems for neutron fission and diffusion, J. Math. Anal. Appl. 120 (1986), 655-669.
134. A. Leung and G. S. Chen, Optimal control of multigroup neutron fission systems, Appl. Math. Optim. 40 (1999), 39-60.
135. A. Leung and G. S. Chen, Existence and global bounds for a fluid model of plasma display technology, J. Math. Anal. Appl. 310 (2005), 436-458.
136. A. Leung and G. Fan, Existence of positive solutions for elliptic systems- degenerate and nondegenerate ecological models, J. Math. Anal. Appl. 151 (1990), 512-531.
137. A. Leung, X. Hou and Y. Li, Exclusive traveling waves for competitive reaction-diffusion systems and their stabilities, J. Math. Anal. Appl. 338 (2008), 902-924.
138. A. Leung and L. Ortega, Bifurcating solutions and stabilities for multigroup neutron fission systems with temperature feedback, J. Math. Anal. Appl. 194 (1995), 489-510.
139. A. Leung and L. Ortega, Bifurcating solutions and stabilities for multigroup neutron fission systems with temperature feedback, J. Math. Anal. Appl. 194 (1995), 489-510.
140. A. Leung and L. Ortega, Positive steady-states for large systems of reaction-diffusion equations: Synthesizing from smaller subsystems, Canadian Appl. Math. Quar. 4 (1996), 175-195.
141. A. Leung and S. Stojanovic, Optimal control for elliptic Volterra-Lotka type equations, J. Math. Anal. Appl. 17 (1993), 603-619.
142. A. Leung and B. Villa, Reaction-diffusion systems for multigroup neutron fission with temperature feedback: Positive steady-state and stability, Diff. Int. Eqs. 10 (1997), 739-756.
143. A. Leung and B. Villa, Bifurcation of reaction-diffusion systems, application to epidemics of many species, J. Math. Anal. Appl. 244 (2000), 542-563.
144. A. Leung and Q. Zhang, Reaction diffusion equations with non-linear boundary conditions, blowup and steady states, Math. Meth. Appl. Sci. 21 (1998), 1593-1617.
145. S. Levin, Models of population dispersal, differential equations and appli-cations, Ecology, Epidemics and Population Problems, edited by S. Busen- berg and K. Cooke, Academic Press, N.Y., 1981.
146. H. Levine and L. Payne, Nonexistence theorems for the heat equations with nonlinear boundary conditions for porous medium equation backward in time, J. Diff. Eqs. 16 (1974), 319-334.
147. J. Lewins, Nuclear Reactor Kinetics and Control, Plenum, New York, 1978.
148. B. Li, H. Weinberger and M. Lewis, Spreading speeds as slowest wave speeds for cooperative systems, Math. Biosci. 196 (2005),82-98.
149. L. Li, Coexistence theorems of steady states for predator-prey interacting systems, Trans. Amer. Math. Soc., 305 (1988), 143-166.
150. L. Li and A. Ghoreishi, On positive solutions of general nonlinear elliptic symbiotic interacting systems, Appl. Anal. 40 (1991), 281-295.
151. L. Li and Y. Liu, Spectral and nonlinear effects in certain elliptic systems of three variables, SIAM J. Math Anal. 24 (1993), 480-498.
152. L. Li and R. Logan, Positive solutions to general elliptic competition models, Diff. Int. Eqs., 4 (1991), 817-834.
153. L. Li and A. Ramm, A singular perturbation result and its applications to mathematical ecology, Proc. Amer. Math. Soc. 111 (1991), 1043-1050.
154. X. Li and J. Yong, Optimal Control Theory for Infinite Dimensional Systems, Birkhauser, Boston, 1995.
155. C. Lin, Z. Lin and W. Jiang, Optimal control of boiling water reactor loading-following operation, Nuclear Sci. Eng. 102 (1989), 134-139.
156. C. Lin, W. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxi system, J. Diff. Eqs. 72 (1988), 1-27.
157. D. Linthicum and N. Farid, Anti-idiotypes, Receptors, and Molecular Mimicry, Springer-Verlag, New York, 1988.
158. J. L. Lions, Equations Differentielles Operationnelles, et problemes aux Limites, Springer-Verlag, Berlin 1961.
159. J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, Berlin, 1972.
160. J. L. Lions and E. Magenes, Nonhomogeneous Boundary Value Problems and Applications I, Springer-Verlag, New York, 1972.
161. Y. Liu, Positive solutions to general elliptic systems, Nonlinear Anal. 25 (1995), 229-246.
162. J. Lopez-Gomez, Spectral Theory and Nonlinear Functional Analysis, Chapman and Hall/CRC Research Notes in Mathematics 426, London, 2001.
163. J. Lopez-Gomez and R. Pardo San Gil, Coexistence regions in Lotka-Volterra models with diffusion, Nonl. Anal. TMA 19 (1992), 11-28.
164. J. Lopez-Gomez and R. Pardo San Gil, Coexistence in simple food chain with diffusion, J. Math. Biology 30 (1992), 655-668.
165. Y. Lou and W. Ni, Diffusion, self-diffusion and cross-diffusion, J. Diff. Eqs., 131 (1996), 79-131.
166. Y. Lou and W. Ni, Diffusion vs cross-diffusion: An elliptic approach, J. Diff. Eqs. 154 (1999), 157-190.
167. R. Lui, Biological growth and spread modeled by systems of recursion, I. mathematical theory, Math. Biosci. 93 (1989), 269-295.
168. H. Matano and M. Mimura, Pattern formation in competition-diffusion systems in non-convex domains, Publ, RIMS. Kyoto Univ. 19 (1983), 1049-1079.
169. P. McKenna and W. Walter, On the Dirichlet problem for elliptic systems, Appl. Anal. 21 (1986), 207-224.
170. R. McOwen, Partial Differential Equations, Methods and Applications, Prentice-Hall, Upper Saddle River, NJ, 1996.
171. M. Mimura, Stationary pattern of some density-dependent diffusion system with competitive dynamics, Hiroshima Math. J. 11 (1981), 621-635.
172. M. Mimura, S. Ei and Q. Fang, Effect of domain-type on the coexistence problems in competition-diffusion system, J. Math. Biol. 29 (1991), 219-237.
173. M. Mimura and K. Kawasaki, Spatial segregation in competitive interaction-diffusion equations, J. Math. Biol. 9 (1980), 49-64.
174. M. Mimura, Y. Nishiura, A. Tesei and T. Tsujikawa, Coexistence problem for two competing species models with density-dependent diffusion, Hiroshima Math. J. 14 (1984), 425-449.
175. K. Mischaikow and V. Hutson, Travelling waves for mutualist species, SIAM J. Math. Anal. 24 (1993), 987-1008.
176. K. Mischaikow, J. Reineck and F. James, Travelling waves in predator-prey systems, SIAM J. Math. Anal. 24 (1993), 1179-1214.
177. X. Mora, Semilinear parabolic problems define semiflows on Ck spaces, Trans. Am. Math. Soc. 278 (1983), 21-55.
178. J. Murray, Mathematical Biology II: Spatial Models and Biomedical Applications, Springer-Verlag, New York, 2003.
179. R. Nussbaum, The fixed point index for locally condensing maps, Ann. Mat. Pura Appl. 87 (1971), 217-258.
180. A. Okubo and S. Levin, Diffusion and Ecological Problems: Modern Perspectives, 2nd ed., Springer-Verlag, Berlin, 2001.
181. L. S. Ortega, On the Leung-Chen feedback model for nuclear fission, Nonlinear Anal. TMA 18 (1992), 353-360.
182. R. Paine, Food web complexity and species diversity, Amer. Natur. 100 (1966), 65-75.
183. C. V. Pao, Coexistence and stability of a competition-diffusion system in population dynamics, J. Math. Anal. Appl., 83 (1981), 54-76.
184. C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992.
185. A. Pazy, Semigroup of Linear Operators and Applications to Partial Dif-ferential Equations, Springer Verlag, N.Y., 1983.
186. L. Peletier, The porous medium equations, in “Applications of Nonlinear Analysis in Physical Science” (H. Amann, N. Bazley, and Kirchgassner Eds), 229-241, London, 1981.
187. M. Pozio and A. Tesei, Degenerate parabolic problems in population dynamics, Japan J. Appl. Math. 2 (1985), 351-380.
188. M. Pozio and A. Tesei, Support properties of solutions for a class of degen-erate parabolic equations, Comm. Partial Differential Equations 12 (1987), 47-75.
189. M. Protter and H. Weinberger, Maximum Principles in Differential Equations, Prentice Hall, Englewood Cliff, N.Y. 1967.
190. P. Quittner, On global existence and stationary solutions for two classes of semilinear parabolic problems, Comment Math. Univ. Carolin 34 (1993), 105-124.
191. P. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal., 1 (1971), 487-513.
192. S. Raul and M. Kushner, Dynamics of coplanar-electrode and plasma display panel cell, I. Basic operation, J. Appl. Phys. 85 (1999), 3460-3469.
193. R. Redheffer and Z. Zhou, Global asymptotic stability for a class of many- variable Volterra prey-predator systems, J. Nonlinear Anal. 5 (1981), 1309-1329.
194. I. Roitt, J. Brostoff and D. Male, Immunology, 2nd ed., Mosby Gower Med. Publishing, 1989.
195. W. Ruan and W. Feng, On fixed point index and multiple steady-state solutions of reaction-diffusion systems, Differential Integral Eqs. 8 (1995), 371-391.
196. W. Ruan and C. Pao, Positive steady-state solutions of competing reaction diffusion system, J. Diff. Eqs. 117 (1995), 411-427.
197. K. Ryu and I. Ahn, Coexistence theorem for steady states for nonlinear self-cross diffusion systems with competitive dynamics, J. Math. Anal. Appl. 283 (2003), 46-65.
198. K. Ryu and I. Ahn, Positive solutions for ratio-dependent predator-prey interaction systems, J. Diff. Eqs. 218 (2005), 117-135.
199. K. Ryu and I. Ahn, Positive solutions of certain elliptic systems with self-diffusions: Non-degenerate vs. degenerate diffusions, Nonl. Anal. 63 (2005), 247-259.
200. M. Salvatori and E. Vitillaro, Decay for the solutions of nonlinear abstract damped evolution equations with applications to partial and ordinary differential systems, Diff. Int. Eqs. 11 (1998), 223-262.
201. D. Sattinger, Monotone methods in nonlinear elliptic and parabolic boundary value problems, Indiana Univ. Math. J. 21 (1972), 979-1000.
202. D. Sattinger, Topics in Stability and Bifurcation Theory, Lecture Notes in Mathematics, Springer Verlag, N.Y., 1973.
203. A. Selvadurai, Partial Differential Equations in Mechanics 1: Fundamentals, Laplace’s Equation, Diffusion, Wave Equation, Springer-Verlag, N.Y., 2000.
204. H. Schaefer, Topological Vector Spaces, Macmillan, New York, 1966.
205. H. Schatzman, Stationary solutions and asymptotic behavior of quasilinear degenerate parabolic equation, Indiana Univ. Math. J. 33 (1984), 1-30.
206. J. Schwartz, Nonlinear Functional Analysis, Gordan and Breach, New York, 1968.
207. N. Shigesada, K. Kawasaki and E. Teramoto, Spatial segregation of inter-acting species, J. Theoret. Biol. 79 (1979), 83-99.
208. D. Simberloff, Experimental zoogeorgraphy of islands: Effects of island size, Ecology 57 (1976), 629-648.
209. H. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Coooperative Systems, Am. Math. Soc., Providence, R.I., 1995.
210. J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, New York, 1983.
211. R. Sperb, Maximum Principles and Their Applications, Academic Press, New York, 1981.
212. B. Stewart, Generation of analytic semigroups by strongly elliptic operators, Trans. Amer. Math. Soc. 199 (1974), 141-161.
213. B. Stewart, Generation of analytic semigroups by strongly elliptic operators under general boundary conditions, Trans. Amer. Math. Soc. 259 (1980), 299-310.
214. S. Stojanovic, Optimal damping control and nonlinear parabolic systems, Numer. Funct. Anal. Optim. 10 (1989), 573-591.
215. W. Strauss, Nonlinear Wave Equations, CBMS No. 71, Amer. Math. Soc., 1989.
216. M. Tang and P. Fife, Propagating fronts for competing species with diffusion, Arch. Rational Mech. Anal. 73 (1978), 69-77.
217. R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, SIAM, Philadelphia, 1983.
218. R. Temam, Infinite Dimensional Dynamical Systems to Mechanics and Physics, Appl. Math. Sciences 68, Springer-Verlag, N.Y., 1997.
219. B. Terney and D. Wade, Optimal control applications in nuclear reactor design and operation, Adv. Nuclear Sci. Tech. 10 (1997).
220. W. Thompsom, An Introduction to Plasma Physics, Addison-Wesley, London, 1964.
221. A. Tihononov, Systems of differential equations containing small parameters in the derivatives, Mat. Sbornik 27 (1952), 147-156 (in Russian).
222. L.Y. Tsai, Existence of solutions of nonlinear elliptic systems, Bulletin Inst. Math. Academia Sinica, 8 (1980), 111-127.
223. M. Vainberg, Variational Methods for the Study of Nonlinear Operators, Holden-Day Series in Mathematical Physics, San Francisco, 1964.
224. G. Veronis and U. Inan, Simulation studies of coplanar electrode and other plasma display panel cell designs, J. Appl. Phys. 91 (2002), 9502-9512.
225. I. Volpert, V. Volpert and V. Volpert, Traveling Wave Solutions of Parabolic Systems, Translations of Math. Mono., Vol. 140, Am. Math. Soc., Providence, R.I., 1994.
226. W. Walter, On the existence and nonexistence in the large of solutions of parabolic differential equations with nonlinear boundary conditions, SIAM J. Math. Anal. 6 (1975), 85-90.
227. M. Wang, Stationary patterns of strongly coupled prey-predator models, J. Math. Anal. Appl., 292 (2004), 484-505.
228. M. Wang and Y. Wu, Global exist and blow up problems for quasi-linear parabolic equations with nonlinear boundary conditions, SIAM J. Math. Anal. 24 (1993), 1515-1521.
229. J. Ward and J. King, Mathematical modeling of avasculartumor growth, IMA J. Math. Appl. Med. Bio. 15 (1998), 1-42.
230. W. Wasow, Asymptotic Expansions for Ordinary Differential Equations, 2nd edn, R.E. Krieger Publ., Huntington, NY, 1976.
231. G. Webb, A bifurcation problem for a nonlinear hyperbolic partial differential equation, SIAM J. Math. Anal. 10 (1979), 922-932.
232. S. Williams and P. Chow, Nonlinear reaction-diffusion models, J. Math. Anal. Appl. 62 (1978), 157-169.
233. R. Wong, Asymptotic and Computational Analysis, Lecture Notes in Pure and Applied Mathematics, Marcel Dekker Inc., N.Y. 1990.
234. J. Wu, Theory and Applications of Partial Functional Differential Equations, Applied Mathematical Sciences, Springer-Verlag, N.Y., 1996.
235. J. Wu and X. Zou, Traveling wave fronts of reaction-diffusion systems with delay, J. Dynamics and Diff. Eq. 13 (2001), 651-687.
236. J. Wu and X. Zou, Erratum to “traveling wave fronts of reaction-diffusion systems with delays”, J. Dynamics and Diff. Eq. 2 (2008), 531-533.
237. Y. Wu, Existence of stationary solutions with transition layers for a class of cross-diffusion systems, Proc. Roy. Soc. Edinburgh, Sect. A, 132 (2002), 1493-1511.
238. S. Xu, (Editor), Neurobiochemistry, Shanghai Medical University Press, 1989.
239. Q. Ye and M. Wang, Travelling wave front solutions of Noyes-Field system for Belousov-Zhabotinskii reaction, Nonlin. Anal. TMA, 11 (1987), 1289-1302.
240. T. Yoshizawa, Stability Theory of Liapunov’s Second Method, University of Tokyo Press, 1966.
241. J. Young, The Life of Mammals, Oxford University Press, New York, 1957.
242. E. Zeidler, Nonlinear Functional Analysis and its Applications, Springer- Verlag, Berlin/New York, 1985.
243. E. Zeidler, Applied Functional Analysis: Applications to Mathematical Physics, Applied Mathematical Sciences, Springer-Verlag, N.Y. 1995.
244. X. Zou and J. Wu, Existence of traveling wavefronts in delayed reaction- diffusion system via monotone iteration method, Proc. Amer. Math. Soc. 125 (1997), 2589-2598.