$L^1$-estimates for the higher-order derivatives of solutions to parabolic equations subject to initial values of bounded total variation

Denis R. Akhmetov; Renato Spigler

doi:10.3934/cpaa.2007.6.1051

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2007, Volume 6, Issue 4: 1051-1074. Doi: 10.3934/cpaa.2007.6.1051

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$L^1$-estimates for the higher-order derivatives of solutions to parabolic equations subject to initial values of bounded total variation

Denis R. Akhmetov^1, and
Renato Spigler^2,

1.
Sobolev Institute of Mathematics, 4, Acad. Koptyug prosp., 630090 Novosibirsk
2.
Dipartimento di Matematica, Università “Roma Tre”, 1, Largo S. L. Murialdo, 00146 Rome

Received: December 31, 2006

Revised: May 31, 2007

Published: November 2007

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Abstract

$L^1$-estimates are established for the higher-order derivatives of classical solutions to the homogeneous Cauchy problem for linear second-order one-dimensional parabolic equations of general form. It is required that the initial data is uniformly continuous and of bounded total variation on some given bounded interval. If the latter condition holds on every bounded interval, then uniform $L^1$-estimates can be proved for the higher-order derivatives. In contrast to earlier findings, where the case of bounded initial data with a continuity modulus satisfying a Dini condition was considered, no constraints are imposed to such a continuity modulus in this paper. In particular, the initial data are allowed to be unbounded. Sets of initial data, in general discontinuous, are also considered.

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Mathematics Subject Classification: Primary: 35B05, 35B30, 35K05, 35R05, 35K15; Secondary: 35C15.

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