## On a partial regularity up to the boundary of weak solutions to quasilinear parabolic systems with quadratic growth.(English. Russian original)Zbl 0969.35032

J. Math. Sci., New York 101, No. 5, 3385-3397 (2000); translation from Zap. Nauchn. Semin. POMI 249, 20-39 (1997).
The paper under review is concerned with the following quasilinear parabolic systems in a cylinder $$Q=\Omega \times (0,T)$$ $$(\Omega \subset \mathbb R^{n},n \geq 2)$$:
(1) $$\frac{\partial u^{k}}{\partial t}- \frac{\partial}{\partial x_{\alpha}}( A^{\alpha\beta}_{kl}(x,t,u) \frac{\partial u^{l}}{\partial x_{\beta}}) + b^{k} (x,t,u,\nabla u)=0$$ $$(k= 1, \cdots , N).$$
Here the coefficients $$A^{\alpha \beta}_{kl}$$ are uniformly continuous and bounded on $$\overline{Q} \times \mathbb R^{N}$$ and satisfy the elliptic condition $A^{\alpha \beta}_{kl}(x,t,u)\zeta^{k}_{\alpha}\zeta^{l}_{\beta}\geq \nu |\zeta |^{2} \qquad \forall \zeta \in \mathbb R^{nN} \qquad (\nu \text{ const } > 0) ,$ and $$b^{k}$$ are Carathéodory functions on $$Q \times \mathbb R^{N} \times \mathbb R^{nN}$$ and satisfy the growth condition $|b (x,t,u,\zeta)|\leq a|\zeta|^{2}+L\quad \forall (x,t,u,\zeta)\in Q \times \mathbb R^{N} \times \mathbb R^{nN}$ ($$a$$= const $$\geq 0$$, $$L=$$ const $$> 0$$).
The main result of the paper is as follows. Let $$\partial \Omega \in C^{2}$$. Let $$u \in W^{1,1}_{2}(Q; \mathbb R^{N})\cap L^{\infty}(Q; \mathbb R^{N})$$ be a weak solution to (1) under Dirichlet or Neumann boundary conditions on $$\partial \Omega \times (0,T)$$. Assume that $$2a||u||_{L^{\infty}(Q; \mathbb R^{N})}< \nu$$. Then there exists an open set $$Q' \subset\overline{Q} \times (0,T)$$ such that $$u$$ is Hölder continuous in $$Q'$$ and the $$(n-\varepsilon)$$-dimensional Hausdorff measure of $$(\overline{\Omega} \times (0,T))\setminus Q'$$ is zero.

### MSC:

 35D10 Regularity of generalized solutions of PDE (MSC2000) 35K50 Systems of parabolic equations, boundary value problems (MSC2000) 35K55 Nonlinear parabolic equations
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### References:

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