The regularity theory for $(m,l)$-Laplacian parabolic equation

A. V. Ivanov

Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 28, Tome 243 (1997), pp. 87-110

Résumé

We present results on regularity for generalized solutions of equations of the form \begin{equation} u_t-\operatorname{div}\{|u|^l|\nabla u|^{m-l}\nabla u\}=0, \quad m>1, \quad l>1-m, \tag{1} \end{equation} obtained recently by the author. We prove a local $L_\infty$ estimate for generalized solutions of this equation (1) under the following condition on the parameters $m$, $l$: \begin{equation} \frac{\sigma+1}{\sigma+2}>\frac1m-\frac1n, \quad \sigma=\frac l{m-1}, \quad m>1, \quad l>1-m. \tag{2} \end{equation} This condition was found by the author is a previous paper (Zapiski Nauchnykh Seminarov POMI, vol. 221, 83–113 (1995)). It was shown there that this condition is necessary for local boundedness of a generalized solution.