**Speaker**

Jihoon Ok, Sogang University; Bianca Stroffolini, University Federico II, Naples

**Abstract**

We consider parabolic systems of the form

\(u_t – \mathrm{div} A(Du) =0 \quad \text{in }\ \Omega_T=\Omega\times(0,T],\)

where \(u:\Omega_T\to \mathbb{R}^N\), \(u=u(x,t)\), is a vector valued function and the nonlinearity \(A:\mathbb{R}^{nN}\to \mathbb{R}^{nN}\) satisfies a general Orlicz growth condition characterized by exponents \(p\) and \(q\), subject to the inequality \(\frac{2n}{n+2}\)

This talk focuses on presenting recent developments in the realm of regularity results concerning the spatial gradient \(Du\) of solutions of the above system, especially, the Hölder continuity when \(A(\xi)\) satisfies the Uhlenbeck structure, i.e., \(A(\xi)=\frac{\varphi'(|\xi|)}{|\xi|}\xi\), and partial Hölder continuity.

These results are joint works with Giovanni Scilla from University of Naples Federico II.