Global existence for a class of highly degenerate parabolic systems

  • Herbert Amann


We consider a system of differential equations describing diffusion in polymers. By means of techniques from semigroup theory we prove that it possesses a unique global smooth solution, thus settling some questions, which came up in numerical investigations of this model. From the mathematical point of view the interest in this problem lies in the fact, that it is a degenerate parabolic (2 × 2)-system, whose “diffusion matrix” has rank one only.

Key words

semilinear parabolic equations analytic semigroups degenerate problems global existence 


  1. [1]
    H. Amann, Periodic solutions of semilinear parabolic evolution equations. Nonlinear Analysis, A Collection of Papers in Honor of Erich H. Rothe (eds. L. Cesari, R. Kanan, H.F. Weinberger), Academic Press, New York: 1978, 1–29.Google Scholar
  2. [2]
    H. Amann, Existence and regularity for semilinear parabolic evolution equations. Ann. Scuola Norm. Sup. Pisa, Ser. IV,XI (1984), 593–676.MathSciNetGoogle Scholar
  3. [3]
    H. Amann, Semigroups and nonlinear evolution equations. Linear Algebra Appl.,84 (1986), 3–32.zbMATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    H. Amann, Dynamic theory of quasilinear parabolic equations. II. Reaction-diffusion systems. Differential-Integral Equations,3 (1990), 13–75.zbMATHMathSciNetGoogle Scholar
  5. [5]
    J. Bergh and J. Löfström, Interpolation Spaces. An Introduction. Springer Verlag, Berlin, 1976.zbMATHGoogle Scholar
  6. [6]
    D.S. Cohen and A.B. White, Jr., Sharp fronts due to diffusion and stress at the glass transition in polymers. J. Polymer Sci., Part B: Polymer Phys.,27 (1989), 1731–1747.CrossRefGoogle Scholar
  7. [7]
    D.S. Cohen and A.B. White, Jr., Sharp fronts due to diffusion and viscoelastic relaxation in polymers. SIAM J. Appl. Math., to appear.Google Scholar
  8. [8]
    R. Nagel, Towards a “matrix theory” for unbounded operator matrices. Math. Z.,201 (1989), 57–68.zbMATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    R. Seeley, Interpolation inL p with boundary conditions. Studia Math.,XLIV (1972), 47–60.MathSciNetGoogle Scholar
  10. [10]
    H. Triebel, Interpolation Theory, Function Spaces, Differential Operators. North Holland, Amsterdam, 1978.Google Scholar

Copyright information

© JJIAM Publishing Committee 1991

Authors and Affiliations

  • Herbert Amann
    • 1
  1. 1.Mathematisches InstitutUniverstät ZürichZürichSwitzerland

Personalised recommendations