Abstract
We prove that some conditions are sufficient for regions to be invariant with respect to strongly coupled quasilinear parabolic systems indivergence form. This result can be applied to certain two population systems where we can compute the boundaries of the invariant regions by solving ordinary differential equations. Under simple conditions on the parameters we get bounded invariant regions.
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References
H. AMANN, Dynamic Theory of Quasilinear Parabolic Equations, I. Abstract Evolution Equations,Nonlinear Anal., Theory Methods Appl. 12, 895–919 (1988)
H. AMANN, Dynamic Theory of Quasilinear Parabolic Equations, II. Reaction-Diffusion Systems,Differ. Integral Equations 3, 13–75 (1990)
H. AMANN, Dynamic Theory of Quasilinear Parabolic Equations, III. Global Existence,Math. Z. 202, 219–250 (1989)
H. AMANN, Quasilinear Evolution Equations and Parabolic Systems,Trans. Amer. Math. Soc. 293, 191–227 (1986)
H. AMANN, Global Existence for Semilinear Parabolic Systems,J. Reine Angew. Math. 360, 47–83 (1985)
H. AMANN, Existence and Regularity for Semilinear Parabolic Evolution Equations,Annali Scuola Norm. Pisa Cl. Sci. (4) 11, 593–676 (1984)
H. AMANN, Global Existence and Positivity for Triangular Quasilinear Reaction-Diffusion Systems,Unpublished Manuscript (1989)
K.N. CHUEH, C.C. CONLEY, J.A. SMOLLER, Positively Invariant Regions for Systems of Nonlinear Diffusion Equations,Indi. Univ. Math. J. 26, 373–392 (1977)
P. DEURING, An Initial-Boundary-Value Problem for a Certain Density-Dependent Diffusion System,Math. Z. 194, 375–396 (1987)
J.U. KIM, Smooth Solutions to a Quasilinear System of Diffusion Equations for a Certain Population Model,Nonlinear Anal., Theory Methods Appl. 8, 1121–1144 (1984)
K.H.W. KÜFNER, Maximumprinzip, Invariante Gebiete und A Priori Schranken für Parabolische Gleichungen,Diploma Thesis, Bayreuth (1990)
K.H.W. KÜFNER, Invariante Gebiete für stark gekoppelte quasilineare parabolische Systeme in Divergenzform und globale Existenz für ein spezielles Gleichungssystem aus der Populationsdynamik,PhD Thesis, Bayreuth (1994)
M. MIMURA, Stationary Pattern of Some Density-Dependent Diffusion System with Competitive Dynamics,Hiroshima Math. J. 11, 621–635 (1981)
M.A. POZIO, A. TESEI, Global Existence of Solutions for a Strongly Coupled Quasilinear Parabolic System,Nonlinear Anal., Theory Methods Appl. 14, 657–689 (1990)
M.H. PROTTER, H.F. WEINBERGER, Maximum Principles in Differential Equations.Springer Verlag, New York (1984). (First published inPrentice Hall, Englewood Cliffs, New Jersey (1967))
R. REDHEFFER, W. WALTER, Invariant Sets for Systems of Partial Differential Equations, I. Parabolic Equations.Arch. Rat. Mech. Anal. 67, 41–52 (1977)
R. REDLINGER, Invariant Sets for Strongly Coupled Reaction-Diffusions Systems under General Boundary Conditions,Arch. Rat. Mech. Anal. 108, 281–291 (1989)
B.J. SCHMITT,Personal Communication.
N. SHIGESADA, K. KAWASAKI, E. TERAMOTO, Spatial Segregation of Interacting Species,J. Theor. Biol. 79, 83–99 (1979)
M. WIEGNER, Global Solutions to a Class of Strongly Coupled Parabolic Systems,Math. Ann. 292, 711–727 (1992)
A. YAGI, Global Solutions to some Quasilinear Parabolic Systems in Population Dynamics,Nonlinear Anal., Theory Methods Appl. 21, 603–630 (1993)
Y. YAMADA, Global Solutions for Quasilinear Parabolic Systems with Cross-Diffusion Effects, Preprint
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Küfner, K.H.W. Invariant regions for quasilinear reaction-diffusion systems and applications to a two population model. NoDEA 3, 421–444 (1996). https://doi.org/10.1007/BF01193829
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DOI: https://doi.org/10.1007/BF01193829