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$W^{2,\infty}$ regularizing effect in a nonlinear, degenerate parabolic equation in one space dimension


Author: Espen Robstad Jakobsen
Journal: Proc. Amer. Math. Soc. 132 (2004), 3203-3213
MSC (2000): Primary 35D10, 35B65; Secondary 35K65, 35K55, 35B37, 49L25
DOI: https://doi.org/10.1090/S0002-9939-04-07577-X
Published electronically: June 16, 2004
MathSciNet review: 2073294
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Abstract: In this paper we provide and analyze a nonlinear degenerate parabolic equation in one space dimension with the following smoothing property: If the initial data is only uniformly continuous, at positive times, the solution has bounded second derivatives (it belongs to $W^{2,\infty}$). We call this surprising phenomenon a $W^{2,\infty}$regularizing effect. So far, such phenomena have only been observed in uniformly parabolic equations.


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Additional Information

Espen Robstad Jakobsen
Affiliation: Department of Mathematical Sciences, Norwegian University of Science and Technology, 7491 Trondheim, Norway
Email: erj@math.ntnu.no

DOI: https://doi.org/10.1090/S0002-9939-04-07577-X
Keywords: Degenerate parabolic equations, Hamilton-Jacobi-Bellman equations, viscosity solutions, regularizing effects, regularity
Received by editor(s): September 12, 2002
Published electronically: June 16, 2004
Communicated by: David S. Tartakoff
Article copyright: © Copyright 2004 American Mathematical Society

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