Regularity theory for the \((m,l)\)-Laplacian parabolic equation (Q1129788)

This interesting paper surveys the state of art in 1996 concerning the regularity theory for doubly degenerate parabolic equations of the type \[ u_t-\text{div}\bigl(u^l |\nabla u|^{m-2} \nabla u\bigr)=0 \] with \(m>1\), \(l>1-m\). The author has contributed a lot of (sharp) results in this context by his own; see also the results of \textit{V. Vespri} [e.g. Manuscr. Math. 75, No. 1, 65-80 (1992; Zbl 0755.35053)]. Let \(\sigma=l/(m-1)\). Starting with nonnegative weak solutions with \(u\in C([0,T]\), \(L_{\sigma+2} (\Omega))\), \(\nabla(u^{\sigma+ 1})\in L_m(\Omega \times(0,T))\), the first theorem of the author proves local \(L_\infty\)-bounds under the sharp assumption \({\sigma+ 1\over \sigma+2} >{1\over m}-{1\over n}\). If one starts with a bounded weak solution, local Hölder estimates may be derived for \(m>1\), \(l\geq 0\). In general this regularity is sharp; but for \(l<1-\varepsilon^* (m,n)\) even local gradient bounds are possible. The last chapter contains a complete proof for the local \(L_\infty\)-estimates of the solution \(u\).