## Partial regularity for subquadratic parabolic systems by $$\mathcal{A}$$-caloric approximation.(English)Zbl 1235.35061

The paper is concerned with nonsingular parabolic systems of the form $$\partial_tu-\nabla\cdot a(t,x,u,Du)=B(x,t,u,Du)$$. The structure function $$a$$ is assumed to fulfill ellipticity and regularity conditions which are fairly general and quite natural. Moreover, it is assumed to have “polynomial” growth with growth rate $$p\in( {2n\over n+1},2)$$. The assumptions on $$B$$ are the usual ones.
Partial regularity of weak solutions is proven, up to a singular set of vanishing measure in spacetime. Different characterizations of the singular sets are given. In the case $$p>2$$ the problem has been solved (see e.g. [F. Duzaar, G. Mingione and K. Steffen, Mem. Am. Math. Soc. 1005, i-x, 118 p. (2011; Zbl 1238.35001)] and the references therein), and even under more general assumptions. Regularity in the subquadratic case $$p<2$$, however, has not been treated before the current paper, except for very special equations. The proof uses a variant of the technique of harmonic approximation.

### MSC:

 35B65 Smoothness and regularity of solutions to PDEs 35K40 Second-order parabolic systems 35K59 Quasilinear parabolic equations

Zbl 1238.35001
Full Text:

### References:

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