Regular points of generalized solutions of nonlinear parabolic systems of higher order (Q1105118)

The authors consider the system \[ (1)\quad \partial u^ k/\partial t+\sum_{| \alpha | \leq m}(-1)^{\alpha} D^{\alpha}_ x a^ k_{\alpha}(x,t,u\quad, ..., D^ m_ x u)=0,\quad k=1,...,N, \] in the space \(V^{m,0}_{q,2}(Q_ T)\) \((Q_ T=[0,T]\times \Omega\), \(\Omega\subset R^ n)\). If \(u(x,t)=(u\) 1(x,t),...,u N(x,t)), \(u^ k(x,t)\in V^{m,0}_{q,2} (Q_ T)\), \(k=1,...,n\), \((x_ 0,t_ 0)\in Q_ T\) is an interior point and \(0<r<r(x_ 0,t_ 0)=\min \{t_ 0^{m},(T-t_ 0)^{m}\), \(dist(x_ 0,\partial \Omega)\}\) one denotes \[ J^ 1_ r[u(x,t),Q_ r(x_ 0,t_ 0)]= \int^{t_ 0+r^{2m}}_{t_ 0-r^{2m}} \int_{B_ r(x_ 0)} (1+\sum^{N}_{\ell =1}\sum_{| \beta | \leq m}| D^{\beta}_ x u^{\ell} |)^{q-2}\cdot \] \[ \cdot \sum^{N}_{k=\ell}\sum_{| \alpha | \leq n}r^{2(| \alpha | -m)}| D^{\alpha}_ x u^ k|^ 2 dx dt, \] where \(Q_ r(x_ 0,t_ 0)=B_ r(x_ 0)\times [t_ 0-r^{2m},t_ 0+r^{2m}]\) and \(B_ r(x_ 0)\) is the sphere of radius r and center \(x_ 0.\) An interior point \((x_ 0,t_ 0)\in Q_ T\) is a regular point of u(x,t) if there exist \(p_{0,\alpha}=(p^ 1_{0,\alpha}, ..., p^ N_{0,\alpha})\), \(0\leq | \alpha | \leq n\), so that \[ \lim_{r\to 0}(1/mes Q_ r(x_ 0,t_ 0))\cdot J_ r[u(x,t)- \sum_{| \alpha | \leq m}p_{0,\alpha}(x-x_ 0)^{\alpha})/\alpha!,Q_ r(x_ 0,t_ 0)]=0. \] The principal result of this paper is the following: Theorem. Let u(x,t) be a generalized solution of the system (1) with \(\partial u/\partial t\in L_ 2(Q_ T)\). If the coefficients of the system (1) satisfy suitable conditions then almost every point of \(Q_ T\) is a regular point for \(u(x,t)\).