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Transactions of the American Mathematical Society

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



On the Cauchy problem and initial traces for a degenerate parabolic equation

Authors: E. DiBenedetto and M. A. Herrero
Journal: Trans. Amer. Math. Soc. 314 (1989), 187-224
MSC: Primary 35K55; Secondary 35K65
MathSciNet review: 962278
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Abstract: We consider the Cauchy problem (f)

$\displaystyle \left\{ {\begin{array}{*{20}{c}} {{u_t} - \operatorname{div}(\ver... ...= {u_0}(x),} \hfill & {x \in {{\mathbf{R}}^N},} \hfill \\ \end{array} } \right.$

and discuss existence of solutions in some strip $ {S_T} \equiv {{\mathbf{R}}^N} \times (0,T)$, $ 0 < T \leq \infty $, in terms of the behavior of $ x \to {u_0}(x)$ as $ \vert x\vert \to \infty $. The results obtained are optimal in the class of nonnegative locally bounded solutions, for which a Harnack-type inequality holds. Uniqueness is shown under the assumption that the initial values are taken in the sense of $ L_{{\text{loc}}}^1({{\mathbf{R}}^N})$.

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Keywords: Nonlinear degenerate parabolic equations, optimal conditions for existence, Harnack inequality
Article copyright: © Copyright 1989 American Mathematical Society