Summary
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.
Article PDF
Similar content being viewed by others
References
S. Agmon,Lectures on Elliptic Boundary Value Problems, Van Nostrand, New York, 1965.
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.
J. R. Cannon - R. E. Ewing,A coupled non-linear hyperbolic-parabolic system, J. Math. Anal. and Appl. (to appear).
R. Carroll,Abstract Methods in Partial Differential Equations, Harper and Row, New York, 1969.
R. Courant -D. Hilbert,Methods of Mathematical Physics, 2 vols., Wiley and Sons, New York, 1962.
R. Courant -P. Lax,On non-linear partial differential equations with two independent variables, Comm. Pure and Appl. Math.,2 (1949), pp. 255–273.
P. Garabedian,Partial Differential Equations, Wiley and Sons, New York, 1964.
W. Hurewicz,Lectures on Ordinary Differential Equations, The M.I.T. Press, Cambridge, Mass., 1958.
I. G. Petrovskii,Partial Differential Equations, W. B. Saunders Company, Philadelphia, Pennsylvania, 1967.
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.
R. E. Showalter,Weak solutions of non-linear evolution equations of Sobolev-Galpern type, J. of Diff. Eqns.,11 (1972), pp. 252–265.
R. E. Showalter -T. W. Ting,Pseudo-parabolic partial differential equations, SIAM J. Math. Anal.,1 (1970), pp. 1–26.
Author information
Authors and Affiliations
Additional information
Entrata in Redazione il 28 luglio 1976.
Rights and permissions
About this article
Cite this article
Ewing, R.E. A coupled non-linear hyperbolic-sobolev system. Annali di Matematica 114, 331–349 (1977). https://doi.org/10.1007/BF02413794
Issue Date:
DOI: https://doi.org/10.1007/BF02413794