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

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December 2007, 6(4): 1051-1074. doi: 10.3934/cpaa.2007.6.1051

$L^1$-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, Russian Federation
2.	Dipartimento di Matematica, Università “Roma Tre”, 1, Largo S. L. Murialdo, 00146 Rome, Italy

Received January 2007 Revised June 2007 Published September 2007

Abstract
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$L^1$-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.

Keywords: initial values of bounded total variation, heat equation, qualitative theory, Cauchy problems for linear parabolic equations, classical solutions, $L^1$-estimates for higher-order derivatives., discontinuous initial data, uniformly continuous initial data.

Mathematics Subject Classification: Primary: 35B05, 35B30, 35K05, 35R05, 35K15; Secondary: 35C1.

Citation: Denis R. Akhmetov, Renato Spigler. $L^1$-estimates for the higher-order derivatives of solutions to parabolic equations subject to initial values of bounded total variation. Communications on Pure & Applied Analysis, 2007, 6 (4) : 1051-1074. doi: 10.3934/cpaa.2007.6.1051

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