Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

A priori estimates and precise regularity for parabolic systems with discontinuous data

Pages: 721 - 742, Volume 13, Issue 3, August 2005      doi:10.3934/dcds.2005.13.721

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Dian Palagachev - Department of Mathematics, Polytechnic University of Bari, 4 E. Orabona Str., 70 125 Bari, Italy (email)
Lubomira Softova - Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria (email)

Abstract: We deal with linear parabolic (in the sense of Petrovskii) systems of order $2b$ with discontinuous principal coefficients. A priori estimates in Sobolev and Sobolev--Morrey spaces are proved for the strong solutions by means of potential analysis and boundedness of certain singular integral operators with kernels of mixed homogeneity. As a byproduct, precise characterization of the Morrey, $BMO$ and Hölder regularity is given for the solutions and their derivatives up to order $2b-1.$

Keywords:  Parabolic systems, a priori estimates, Morrey spaces, Hölder regularity.
Mathematics Subject Classification:  35K40 (primary); 35R05, 35B45, 35B65, 42B20,46E35 (secondary).

Received: September 2004;      Revised: March 2005;      Published: May 2005.