\(L^ p\) theory and partial Hölder continuity for quasilinear parabolic systems of higher order with strictly controlled growth
Let \(\Omega\) be a bounded open set in \({\mathbb{R}}^ n\) \((n>2)\) with smooth boundary. \(X=(x,t)\) is a variable in \(Q=\Omega \times (-T,0)\), where \(T>0\). We put \(p\equiv \{p^{\alpha}\}_{| \alpha | \leq m}\), \(p^*\equiv \{p^{\alpha}\}_{| \alpha | \leq m-1}\), \(p^{\alpha}\in {\mathbb{R}}^ n\). Then \(p\in {\mathcal R}\equiv \prod_{| \alpha | \leq m}{\mathbb{R}}^ n_{\alpha}\), \(p^*\in {\mathcal R}^*\equiv \prod_{| \alpha | \leq m-1}{\mathbb{R}}^ n_{\alpha}\). Consider in Q the following quasilinear system \[ (1)\quad (-1)^ m\sum_{| \alpha | =m}\sum_{| \beta | =m}D^{\alpha}(A_{\alpha \beta}(x,\delta u)D^{\beta}u)+\partial u/\partial t=F\equiv \] \[ \equiv (-1)^ m\sum_{| \alpha | =m}D^{\alpha}f^{\alpha}(X,\delta u)+\sum_{| \alpha | \leq m-1}(-1)^{| \alpha |}D^{\alpha}f^{\alpha}(X,Du) \] where \(A_{\alpha \beta}(x,p^*)\), \(| \alpha | =m\), \(| \beta | =m\), are \(n\times n\) matrices, uniformly continuous and bounded in \(\bar Q\times {\mathcal R}^*\) and such that \[ (2)\quad \sum_{| \alpha | =m}\sum_{| \beta | =m}(A_{\alpha \beta}(X,p^*)\xi^{\beta}| \xi^{\alpha})\geq \sum_{| \alpha | =m}\| \xi^{\alpha}\|^ 2,\quad \nu >0, \] for \((X,p^*)\in \bar Q\times {\mathcal R}^*\) and \(\{\xi^{\alpha}\}_{| \alpha | =m}\), \(\xi^{\alpha}\in {\mathbb{R}}^ n\), \(f^{\alpha}(X,p^*)\), \(| \alpha | =m\), and \(f^{\alpha}(X,p)\), \(| \alpha | \leq m-1\), are vectors in \({\mathbb{R}}^ n\), measurable in \(X\in Q\), continuous in \(p^*\in {\mathcal R}^*\) and \(p\in {\mathcal R}\) respectively, which satisfy the conditions: \[ (3)\quad \| f^{\alpha}(X,p^*)\| \leq g^{\alpha}(X)+c\sum_{| \beta | \leq m-1}\| p^{\beta}\|^{Q(m,| \beta |)},\quad | \alpha | =m, \] \[ \| f^{\alpha}(X,p)\| \leq g^{\alpha}(X)+c\sum_{| \beta | \leq m}\| p^{\beta}\|^{Q(| \alpha |,| \beta |)},\quad | \alpha | \leq m-1, \] with \[ (4)\quad 1\leq \theta (m,| \beta |)<(n+2m/n+2| \beta |),\quad | \beta | \leq m-1, \] \[ 1\leq \theta (| \alpha |,| \beta |)<(n+4m-2| \alpha | /n+2| \beta |),\quad | \alpha | \leq m-1,\quad | \beta | \leq m, \] \[ (5)\quad g^{\alpha}(X)\in L^ p(Q),\quad | \alpha | =m,\quad g^{\alpha}(X)\in L^{p/{\bar \gamma}| \alpha |}(Q),\quad | \alpha | \leq m-1 \] where \({\bar \gamma}| \alpha | =(n+4m- 2| \alpha | /n+2m)\), \(p>n+2m\). Then, the authors show: if \(u\in L^ 2(-T,0;H^ m(\Omega,{\mathbb{R}}^ n))\cap L^{\infty}(-T,0;L^ 2(\Omega,{\mathbb{R}}^ n))\) is a solution of (1) with these conditions, there exists a set \(Q_ 0\subset Q\), closed in Q, such that \(\mu_{n+2m-2}(Q_ 0)=0\) and, for \(| \alpha | =m-1\), \(D^{\alpha}u\in C^{0,\gamma}(Q\setminus Q_ 0,{\mathbb{R}}^ n),\quad \gamma <1-(n+2m)/p,\) where \(\mu_{m+2m-2}\) is the \((n+2m-2)\)-dimensional Hausdorff measure with respect to the parabolic metric \(\delta(X,Y)\) defined by \(\delta (X,Y)=\max \{\| x-y\|,| t-\tau |^{1/2m}\},\) \(X=(x,t),\) \(Y=(y,\tau).\)
scientific article; zbMATH DE number 3914833