Annali di Matematica Pura ed Applicata

, Volume 114, Issue 1, pp 331–349 | Cite as

A coupled non-linear hyperbolic-sobolev system

  • Richard E. Ewing


A boundary-initial value problem for a quasilinear hyperbolic system in one space variable is coupled to a boundary-initial value problem for a quasilinear equation of Sobolev type in two space variables of the form Mut(x, t)+L(t) u (x, t)=f(x, t, u(x, t)) where M and L(t) are second order elliptic spacial operators. The coupling occurs through one of the boundary conditions for the hyperbolic system and the source term in the equation of Sobolev type. Such a coupling can arise in the consideration of oil flowing in a fissured medium and out of that medium via a pipe. Barenblatt, Zheltov, and Kochina[2] have modeled flow in a fissured medium via a special case of the above equation. A local existence and uniqueness theorem is demonstrated. The proof involves the method of characteristics, some applications of results of R. Showalter and the contraction mapping theorem.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    S. Agmon,Lectures on Elliptic Boundary Value Problems, Van Nostrand, New York, 1965.Google Scholar
  2. [2]
    G. I. Barenblatt -I. P. Zheltov -I. N. Kochina,Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks, J. Appl. Math. Mech.,24 (1960), pp. 1286–1303.CrossRefGoogle Scholar
  3. [3]
    J. R. Cannon - R. E. Ewing,A coupled non-linear hyperbolic-parabolic system, J. Math. Anal. and Appl. (to appear).Google Scholar
  4. [4]
    R. Carroll,Abstract Methods in Partial Differential Equations, Harper and Row, New York, 1969.Google Scholar
  5. [5]
    R. Courant -D. Hilbert,Methods of Mathematical Physics, 2 vols., Wiley and Sons, New York, 1962.Google Scholar
  6. [6]
    R. Courant -P. Lax,On non-linear partial differential equations with two independent variables, Comm. Pure and Appl. Math.,2 (1949), pp. 255–273.MathSciNetGoogle Scholar
  7. [7]
    P. Garabedian,Partial Differential Equations, Wiley and Sons, New York, 1964.Google Scholar
  8. [8]
    W. Hurewicz,Lectures on Ordinary Differential Equations, The M.I.T. Press, Cambridge, Mass., 1958.Google Scholar
  9. [9]
    I. G. Petrovskii,Partial Differential Equations, W. B. Saunders Company, Philadelphia, Pennsylvania, 1967.Google Scholar
  10. [10]
    R. E. Showalter,Existence and representation theorems for a semilinear Sobolev equation in a Banach space, SIAM J. Math. Anal.,3 (1972), pp. 527–543.CrossRefMATHMathSciNetGoogle Scholar
  11. [11]
    R. E. Showalter,Weak solutions of non-linear evolution equations of Sobolev-Galpern type, J. of Diff. Eqns.,11 (1972), pp. 252–265.MATHMathSciNetGoogle Scholar
  12. [12]
    R. E. Showalter -T. W. Ting,Pseudo-parabolic partial differential equations, SIAM J. Math. Anal.,1 (1970), pp. 1–26.CrossRefMathSciNetGoogle Scholar

Copyright information

© Fondazione Annali di Matematica Pura ed Applicata 1977

Authors and Affiliations

  • Richard E. Ewing
    • 1
  1. 1.RochesterU.S.A.

Personalised recommendations