Existence of weak solutions of parabolic systems with \(p,q\)-growth (Q304488)

The author considers evolutionary problems associated with a convex integrand \(f:\Omega_T\times \mathbb R^{N_n}\rightarrow [0,\infty)\), which is \(\alpha\)-Hölder continuous with respect to the \(x\)-variable and satisfies a non-standard \(p,q\)-growth condition. The author proves the existence of weak solutions \(u:\Omega_T\rightarrow \mathbb R^N\), which solve \[ \partial_t u-\operatorname{div}\partial_\zeta f(x,t,Du)=0 \] weakly in \(\Omega_T\). Therefore, the author uses the concept of variational solutions, which exist under a mild assumption on the gap \(q-p\), namely \[ \frac{2n}{n+2}<p\leq q<p+1. \] For \[ \frac{2n}{n+2}<p\leq q<p+\frac{\min\{2,p\}\alpha}{n+2}, \] the author proves that the spatial derivative \(Du\) of a variational solution \(u\) admits a higher integrability and is accordingly a weak solution.