# American Institute of Mathematical Sciences

2003, 2003(Special): 727-733. doi: 10.3934/proc.2003.2003.727

## Parabolic systems with non continuous coefficients

 1 Dipartimento di Matematica e Informatica, Viale Andrea Doria, 6, 95128- Catania, Italy

Received  September 2002 Revised  March 2003 Published  April 2003

In this note we are interested in the local regularity of the highest order derivatives of the solutions of the system

$T u = fi(y)$      $i = 1,...,N where the known terms$f_i$are in Lebesgue spaces and the differential the parabolic operator$T$has the form$ut - \sum_{j=1}^{N}\sum_{|\alpha|=2s} a^(\alpha)_(ij) (y)D^(\alpha) u_j (y) + \sum_{j=1}^{N}\sum_{|\alpha|<=2s-1} b^(\alpha)_(ij) (y)D^(\alpha) u_j (y)$. have discontinuous coefficients. Citation: Maria Alessandra Ragusa. Parabolic systems with non continuous coefficients. Conference Publications, 2003, 2003 (Special) : 727-733. doi: 10.3934/proc.2003.2003.727  [1] Kai Liu. Stationary solutions of neutral stochastic partial differential equations with delays in the highest-order derivatives. Discrete and Continuous Dynamical Systems - B, 2018, 23 (9) : 3915-3934. doi: 10.3934/dcdsb.2018117 [2] Denis R. Akhmetov, Renato Spigler.$L^1$-estimates for the higher-order derivatives of solutions to parabolic equations subject to initial values of bounded total variation. Communications on Pure and Applied Analysis, 2007, 6 (4) : 1051-1074. doi: 10.3934/cpaa.2007.6.1051 [3] Marina V. Plekhanova. Strong solutions of quasilinear equations in Banach spaces not solvable with respect to the highest-order derivative. Discrete and Continuous Dynamical Systems - S, 2016, 9 (3) : 833-846. doi: 10.3934/dcdss.2016031 [4] Jie Jiang. Global stability of Keller–Segel systems in critical Lebesgue spaces. Discrete and Continuous Dynamical Systems, 2020, 40 (1) : 609-634. doi: 10.3934/dcds.2020025 [5] Xuanji Jia, Yong Zhou. Regularity criteria for the 3D MHD equations via partial derivatives. II. Kinetic and Related Models, 2014, 7 (2) : 291-304. doi: 10.3934/krm.2014.7.291 [6] Doyoon Kim, Kyeong-Hun Kim, Kijung Lee. Parabolic Systems with measurable coefficients in weighted Sobolev spaces. Communications on Pure and Applied Analysis, , () : -. doi: 10.3934/cpaa.2022062 [7] Maria Alessandra Ragusa, Atsushi Tachikawa. Estimates of the derivatives of minimizers of a special class of variational integrals. Discrete and Continuous Dynamical Systems, 2011, 31 (4) : 1411-1425. doi: 10.3934/dcds.2011.31.1411 [8] Dian Palagachev, Lubomira Softova. A priori estimates and precise regularity for parabolic systems with discontinuous data. Discrete and Continuous Dynamical Systems, 2005, 13 (3) : 721-742. doi: 10.3934/dcds.2005.13.721 [9] Lijuan Wang, Jun Zou. Error estimates of finite element methods for parameter identifications in elliptic and parabolic systems. Discrete and Continuous Dynamical Systems - B, 2010, 14 (4) : 1641-1670. doi: 10.3934/dcdsb.2010.14.1641 [10] Horst Heck, Matthias Hieber, Kyriakos Stavrakidis.$L^\infty\$-estimates for parabolic systems with VMO-coefficients. Discrete and Continuous Dynamical Systems - S, 2010, 3 (2) : 299-309. doi: 10.3934/dcdss.2010.3.299 [11] Hongjie Dong, Kunrui Wang. Interior and boundary regularity for the Navier-Stokes equations in the critical Lebesgue spaces. Discrete and Continuous Dynamical Systems, 2020, 40 (9) : 5289-5323. doi: 10.3934/dcds.2020228 [12] Julii A. Dubinskii. Complex Neumann type boundary problem and decomposition of Lebesgue spaces. Discrete and Continuous Dynamical Systems, 2004, 10 (1&2) : 201-210. doi: 10.3934/dcds.2004.10.201 [13] Carlo Bardaro, Ilaria Mantellini. Boundedness properties of semi-discrete sampling operators in Mellin–Lebesgue spaces. Mathematical Foundations of Computing, 2022, 5 (3) : 219-229. doi: 10.3934/mfc.2021031 [14] Thuy N. T. Nguyen. Uniform controllability of semidiscrete approximations for parabolic systems in Banach spaces. Discrete and Continuous Dynamical Systems - B, 2015, 20 (2) : 613-640. doi: 10.3934/dcdsb.2015.20.613 [15] Anton Schiela, Julian Ortiz. Second order directional shape derivatives of integrals on submanifolds. Mathematical Control and Related Fields, 2021, 11 (3) : 653-679. doi: 10.3934/mcrf.2021017 [16] Weisheng Niu, Yao Xu. Convergence rates in homogenization of higher-order parabolic systems. Discrete and Continuous Dynamical Systems, 2018, 38 (8) : 4203-4229. doi: 10.3934/dcds.2018183 [17] Elena-Alexandra Melnig. Internal feedback stabilization for parabolic systems coupled in zero or first order terms. Evolution Equations and Control Theory, 2021, 10 (2) : 333-351. doi: 10.3934/eect.2020069 [18] Shaohua Chen. Boundedness and blowup solutions for quasilinear parabolic systems with lower order terms. Communications on Pure and Applied Analysis, 2009, 8 (2) : 587-600. doi: 10.3934/cpaa.2009.8.587 [19] Yuri Latushkin, Valerian Yurov. Stability estimates for semigroups on Banach spaces. Discrete and Continuous Dynamical Systems, 2013, 33 (11&12) : 5203-5216. doi: 10.3934/dcds.2013.33.5203 [20] Nobuyuki Kato, Norio Kikuchi. Campanato-type boundary estimates for Rothe's scheme to parabolic partial differential systems with constant coefficients. Discrete and Continuous Dynamical Systems, 2007, 19 (4) : 737-760. doi: 10.3934/dcds.2007.19.737

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