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Boundary-Value Problems of a Degenerate Sobolev-Type Differential Equation

Published online by Cambridge University Press:  20 November 2018

C. V. Pao
C. V. Pao*
Affiliation:
Dept. of Math, North Carolina State University, RaleighNorth Carolina 27607 U.S.A.
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Abstract

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The purpose of this paper is to study a degenerate Sobolev type partial differential equation in the form of Mut + Lu = f, where M and L are second order partial differential operators defined in a domain (0, T]×Ω in Rn+1. The degenerate property of the equation is in the sense that both M and L are not necessarily strongly elliptic and their coefficients may vanish or be negative in some part of the domain (0, T]×Ω. Two types of boundary conditions are investigated.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 20 , Issue 2 , 01 June 1977 , pp. 221 - 228
Copyright
Copyright © Canadian Mathematical Society 1977

References

1. Barenblat, G., Zheltor, I., and Kochiva, I., Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks, J. Appl. Math. Mech., 24 (1960), 1286-1303.Google Scholar
2. Chen, P. and Gurtin, M., On a theory of heat conduction involving two temperatures, Z. Angew. Math. Phys., 19 (1968), 614-627.Google Scholar
3. Ford, W. T., The first initial boundary-value problem for a non-uniform parabolic equation, J. Math. Anal. Appl., 40 (1972), 131-137.Google Scholar
4. Ivanov, A. V.,A boundary value problem for degenerate second order parabolic linear equations (Russian), Zap. Nauch. Sem. Leningrad Otdel. Math. Inst. Steklov., 14 (1969), 48-88.Google Scholar
5. Lagnese, J., General boundary value problems for differential equations of Sobolev-Galpern type, SIAM J. Math. Anal., 3 (1972), 105-119.Google Scholar
6. Lions, J. L., Equations Différentielles Operationelles, Springer-Verlag, Berlin, 1961.Google Scholar
7. Oleinnik, O. A., On the smoothness of the solutions of degenerate elliptic and parabolic equations, Soviet Math. Dokl., 6 (1965), 972-976.Google Scholar
8. Pao, C. V., On a non-uniform parabolic equation with mixed boundary condition, Proc. Amer. Math. Soc., 49 (1975), 83-89.Google Scholar
9. Rao, V. R. Gopala and Ting, T. W., Solutions of pseudo-heat equations in whole space, Arch. Rational Mech. Anal., 49 (1972), 57-78.Google Scholar
10. Showalter, R. E., Degenerate evolution equations and applications, Indiana Univ. Math. J., 23 (1974), 655-677.Google Scholar
11. Sobolev, V. V., A Treatise on Radiative Transfer, Van Nostrand, New York, 1963.Google Scholar
12. Ting, T. W., Certain non-study flows of second order fluids, Arch. Rational Mech. Anal., 14 (1963), 1-26.Google Scholar