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Properties of Solutions of Parabolic Equations and Inequalities

Published online by Cambridge University Press:  20 November 2018

M. H. Protter
M. H. Protter*
Affiliation:
University of California, Berkeley
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In this paper we shall be concerned with two problems: (i) the asymptotic behavior of solutions of parabolic inequalities and (ii) the uniqueness of the Cauchy problem for such inequalities when the data are prescribed on a portion of a time-like surface. The unifying feature of these rather separate problems is the employment of integral estimates of the same type in both cases.

We consider parabolic operators in self-adjoint form

(1)

as well as the non-self-adjoint operator

(2)

where the coefficients aij(x, t) = aij (x1, x2... , xn, t) are C1 functions of x and t and the bij= bij(x, t) are C2 functions of x and t.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 13 , 1961 , pp. 331 - 345
Copyright
Copyright © Canadian Mathematical Society 1961

References

1. Lax, P. D., A stability theorem for solutions of abstract differential equations, and its application to the study of the local behavior of elliptic equations, Comm. Pure and Applied Math., 9 (1956), 747–766.Google Scholar
2. Lees, M. and Protter, M. H., Unique continuation for parabolic differential equations and inequalities, to appear.Google Scholar
3. Lees, M., Asymptotic behavior of solutions of parabolic differential inequalities, to appear.Google Scholar
4. Mizohata, S., Unicité du prolongement des solutions pour quelques opérateurs différentiels paraboliques, Mem. Coll. Sci. Univ. Kyoto, Ser. Al, 31 (1958), 219–239.Google Scholar
5. Protter, M. H., Unique continuation for elliptic equations, Trans. Amer. Math. Soc, 95 (1960), 81–91.Google Scholar