The VishikLyusternik method in the mixed problem for parabolic operators unresolved with respect to the highest time derivative
Author:
L. R. Volevich
Translated by:
O. A. Khleborodova
Original publication:
Trudy Moskovskogo Matematicheskogo Obshchestva, tom 68 (2007).
Journal:
Trans. Moscow Math. Soc. 2007, 6792
MSC (2000):
Primary 35K40; Secondary 35B25
Published electronically:
November 15, 2007
MathSciNet review:
2429267
Fulltext PDF Free Access
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Abstract: We consider the mixed problem for parabolic operators unresolved with respect to the highest time derivative with boundary conditions of general type and zero initial conditions. We present an analog of the ShapiroLopatinskii condition that allows one to obtain twosided a priori estimates in specially constructed function spaces. In the case considered in this paper the characteristic equation in the halfspace has two groups of roots with different asymptotics. Because of this, the crucial role in the study of the problem is played by the VishikLyusternik method in the form presented by Volevich (2006).
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 L. S. Frank, Coercive singular perturbations. I. A priori estimates. Annali Mat. Pura Appl. 119 (1979), 41113. MR 551218 (81b:35025)
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 I. Fedotov and L. R. Volevich. The Cauchy problem for hyperbolic equations not resolved with respect to the highest time derivative. Russian J. Math. Phys. 13 (2006), no. 3, 278292. MR 2262830
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Additional Information
DOI:
http://dx.doi.org/10.1090/S0077155407001616
PII:
S 00771554(07)001616
Published electronically:
November 15, 2007
Additional Notes:
The work is supported by the Russian Foundation of Fundamental Research, Grant 060100096.
Dedicated:
To Mark Iosifovich Vishik for his 80th birthday
Article copyright:
© Copyright 2007
American Mathematical Society
