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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Second-order time degenerate parabolic equations
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by Margaret C. Waid PDF
Trans. Amer. Math. Soc. 170 (1972), 31-55 Request permission

Abstract:

We study the degenerate parabolic operator $Lu = \sum \nolimits _{i,j = 1}^n {{a^{ij}}{u_{{x_j}{x_j}}}} + \sum \nolimits _{i = 1}^n {{b^i}{u_{{x_i}}}} - c{u_t} + du$ where the coefficients of $L$ are bounded, real-valued functions defined on a domain $D = \Omega \times (0,T] \subset {R^{n + 1}}$. Classically, $c(x,t) \equiv 1$ or, equivalently, $c(x,t) \geq \eta > 0$ for all $(x,t) \in \bar D$. We assume only that $c$ is non-negative. We prove weak maximum principles and Harnack inequalities. Assume that ${a^{ij}}$ is constant, the coefficients of $L$ and $f$ and their derivatives with respect to time are uniformly Hölder continuous (exponent $\alpha$) in $\bar D,\bar D$ has sufficiently nice boundary, $c > 0$ on the normal boundary of $D$, $\psi \in {\bar C_{z + \alpha }}$, and $L\psi = f$ on $\partial B = \partial (\bar D \cap \{ t = 0\} )$. Then there exists a unique solution $u$ of the first initial-boundary value problem $Lu = f,u = \psi$ on $\bar B + (\partial B \times [0,T])$; and, furthermore, $u \in {\bar C_{2 + \alpha }}$. All results require proofs that differ substantially from the classical ones.
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Additional Information
  • © Copyright 1972 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 170 (1972), 31-55
  • MSC: Primary 35K10
  • DOI: https://doi.org/10.1090/S0002-9947-1972-0304860-1
  • MathSciNet review: 0304860