Zhang Guo-chu. Stability of Stationary State Solution for a Reaction Density-Dependent Diffusion Equation[J]. Applied Mathematics and Mechanics, 1989, 10(11): 987-996.
Citation: Zhang Guo-chu. Stability of Stationary State Solution for a Reaction Density-Dependent Diffusion Equation[J]. Applied Mathematics and Mechanics, 1989, 10(11): 987-996.

Stability of Stationary State Solution for a Reaction Density-Dependent Diffusion Equation

  • Received Date: 1988-11-29
  • Publish Date: 1989-11-15
  • In this paper we are interested in the large time behavior of the nonlinear diffusion equation u1=(φ(u))xx+φ(u), (x∈R, f∈R+=(0,+∞)) We consider functions φ(u) and φ(u) which allow the equation to possess traveling wave solutions. We first present an existence and uniqueness as well as some comparison principle result of generalized solutions to the Cauchy problem. Then we give for some threshold results, from which we can see that u=a is stable, while u=0 or u=1 is unstable under some assumptions, etc.
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  • [1]
    Aronson,D.G.,Density-dependent interaction-diffusion systems,Dynamics and Modelling of Reactive Systems,Academic Press,New York(1980),161-176.
    [2]
    Aronson,D.G.and H.F.Weinberger,Non-linear diffusion in population genetics,combustion,and nerve propogation,Proceedings of the Tulane Program in Partial Differential Equations and Related Topics,Lecture Notes in Mathematics,Springer-Verlag,Berlin(1975),446.
    [3]
    Conway,J.B.Functions of One Complex Variable,Springer-Verlag,New York(1978).
    [4]
    Eife,P.C.,Mathematical Aspects of Reacting and Diffusing Systems,Lecture Notes in Biomathematics,Springer-Verlag,Berlin,Heidelberg,New York(1979).
    [5]
    Friedman,A.,Partial Differential Equations of Parabolic Type,Prentice-Hall,New Jersey(1964).
    [6]
    Gurtin,M.E.and R.C.MacCamy,On the diffusion of biological populations.Math.Biosci.,33,(1977),35-49.
    [7]
    Hale,J.K.,Ordinary Differential Equations,Robert E.Krieger Publishing Company,Florida(1980).
    [8]
    Kalashnikov,A.S.,The propagation of disturbance in problems of non-linear heat conduction with absorption,Zh.Vychisl Mat.Mat.Fiz.,14,4(1974),891-905.
    [9]
    Kalashinikov,A.S.,The Cauchy problem in the class of increasing functions of equations of the non-stationary seepage type,Vesln,Mosk,U_n-t_n,Matem.Mekham,6. (1963),17-23.
    [10]
    Kruzhkov,S.N.,Results concerning the nature of the continuity of the results of parabolic equations and some applications,Matem.Zametki,6,1(1969),97-108;Translations Mathematical Notes,6,1(1969),517-523.
    [11]
    Ladyzhenskaya.O.A.,V.A.Solonnikov,and N.N.Uraltseva,Linear and Quasi-Linear Equations of the Parabolic Type.Nauka,Moscow(1967).
    [12]
    Oleinik,Olga,On some degenerate quasilinear parabolic equations,Conferenze tenute al Sominario di Analisi nei giorni 18,19,Aprile(1963).
    [13]
    Protter,M.H.and H.F.Weinberger,Maximum Principle in Differential Equations,Prentice-Hall.Englewood Cliffs(1967).
    [14]
    Rudin.Walte,Real and Complex Analysis,McGraw,(1973).
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