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

Arkhipova, A. A.

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.

Reviewer: Joachim Naumann (Berlin)

MSC:

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

Keywords:

Dirichlet or Neumann boundary conditions; partial Hölder continuity of weak solutions; Hausdorff measure

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