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On the nonlinear parabolic systems in divergence form. Hölder continuity and partial Hölder continuity of the solutions. (English) Zbl 0704.35024

The author considers second-order nonlinear parabolic systems of the type \[ -\sum^{n}_{i=1}D_ iA^ i(X,u,Du)+\frac{\partial u}{\partial t}=- \sum^{n}_{i=1}D_ iB^ i(X,u)+B^ 0(X,u,Du) \] in the cylinder \(Q=\Omega \times (-T,0)\), whose generic point is \(X=(x,t)\). He supposes that the system is strongly parabolic and that the vectors \(A^ i,B^ i,B^ 0\) have strictly controlled growths. Under some additional hypotheses on \(A^ i\), he proves the main result: When \(n\leq 2\) the solutions are \(\mu\)-partial Hölder continuous in Q for a suitable \(\mu\in (0,1)\) and their singular sets have zero n-dimensional Hausdorff measure; whereas, for any n, the solutions are \(\mu\)-partial Hölder continuous for all \(\mu\in (0,1)\), but, about their singular set \(Q_ 0\) the author can merely say, at present, that the Lebesgue measure of \(Q_ 0\) is zero.
These results are obtained as a particular case of regularity or partial regularity in the \({\mathcal L}^{2,\lambda}(Q,d)\) spaces. It is remarkable that in the theory of the \({\mathcal L}^{2,\lambda}\)-regularity for nonlinear parabolic systems, as developed by the author during the last few years, those systems of the type \(-\sum^{n}_{i=1}D_ iA^ i(Du)+\partial u/\partial t=0\) play an analogous role to that played by linear systems, with constant coefficients and reduced to the principal part, in the theory of linear or quasilinear systems. These systems are intensively studied in the paper, and the results obtained are of value by themselves. Moreover, other results are obtained when the coefficients \(A^ i\) are of the type \(A^ i(X,Du)\).

MSC:

35D10 Regularity of generalized solutions of PDE (MSC2000)
35K55 Nonlinear parabolic equations
35K25 Higher-order parabolic equations
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