The Vishik-Lyusternik 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**, 67-92

MSC (2000):
Primary 35K40; Secondary 35B25

Published electronically:
November 15, 2007

MathSciNet review:
2429267

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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 Shapiro-Lopatinskii condition that allows one to obtain two-sided *a priori* estimates in specially constructed function spaces. In the case considered in this paper the characteristic equation in the half-space has two groups of roots with different asymptotics. Because of this, the crucial role in the study of the problem is played by the Vishik-Lyusternik method in the form presented by Volevich (2006).

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Additional Information

DOI:
https://doi.org/10.1090/S0077-1554-07-00161-6

Published electronically:
November 15, 2007

Additional Notes:
The work is supported by the Russian Foundation of Fundamental Research, Grant 06-01-00096.

Dedicated:
To Mark Iosifovich Vishik for his 80th birthday

Article copyright:
© Copyright 2007
American Mathematical Society