Abstract
In this paper I discuss nonlinear parabolic systems that are generalizations of scalar diffusion equations. More precisely, I consider systems of the form
where \({\Phi(z)}\) is a strictly convex function. I show that when \({\Phi}\) is a function only of the norm of u, then bounded weak solutions of these parabolic systems are everywhere Hölder continuous and thus everywhere smooth. I also show that the method used to prove this result can be easily adopted to simplify the proof of the result due to Wiegner (Math Ann 292(4):711–727, 1992) on everywhere regularity of bounded weak solutions of strongly coupled parabolic systems.
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Acknowledgements
The author would like to thank his thesis advisor L. C. Evans for his continuous support and helpful advise.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Trokhimtchouk, M. Everywhere regularity of certain nonlinear diffusion systems. Calc. Var. 37, 407–422 (2010). https://doi.org/10.1007/s00526-009-0269-7
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DOI: https://doi.org/10.1007/s00526-009-0269-7