Global solvability of the Cauchy-Dirichlet problem for the class of nondiagonal parabolic systems with \(q\)-nonlinearity in the gradient, \(1< q< 2\)
The author considers parabolic systems of the form \[ u_t^k - \frac{d}{dx_{\alpha}}f_{u^k_{x_{\alpha}}}(x,u,\nabla u) + f_{u^k}(x,u,\nabla u)=0 \quad \text{on}\quad Q, \tag{1} \] where \(Q = \Omega \times (0,T)\); \(\Omega\) is a smoothly bounded domain in \(\mathbb R^2\), \(T\) a fixed positive number; \(u:\Omega \to \mathbb R^N\) and \(f\) twice continuously differentiable function such that \(f(.,.,\Theta)\thickapprox \Theta^q \) for a \(q \in (1,2)\), \(f\) satisfies natural growth conditions and an ellipticity condition of the form \[ \langle f_{\Theta,\Theta}(.,.,\Theta) \xi,\xi\rangle \geq \nu (1 + | \Theta| )^{q-2}| \xi| ^2 \] for any \(\Theta, \xi \in \mathbb R^{2N}\) and a fixed positive \(\nu\). No smallness condition is put on the lower order term \( f_{u^k}(x,u,\nabla u)\). Moreover, the initial and boundary conditions \[ u = 0 \quad \text{on}\quad \partial \Omega \times (0,T), u(.,0) = \varphi\quad \text{on}\;\Omega \tag{2} \] are required with a function \(\varphi \in C^{2+\alpha}(\Omega)\) satisfying compatibility conditions on \(\partial \Omega\). Under these assumptions, the author proves that for any \(\alpha \in (0,1)\), \(T>0\) there exists a weak solution \(u \in H^{2+\alpha, 1 + \alpha/2}(Q_0)\) to (1), (2) which is smooth on an open subset \(Q_0 \subset Q\) of full measure. Moreover, \(u \in L^{\infty}((0,T), W^{1,q}(\Omega))\) and \(u_t \in L^2(Q)\). The singular set \(\Sigma = \overline{Q} \setminus Q_0\) has \(q\)-parabolic Hausdorff dimension at most two and for any \(\tau \in (0,T)\) it holds \(\dim_H \Sigma \cap \{[x,\tau]\in Q\} \leq 2-q \).
scientific article; zbMATH DE number 2183339