Maximum modulus estimates for generalized solutions of doubly nonlinear parabolic equations

A. V. Ivanov

Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 26, Tome 221 (1995), pp. 83-113

Résumé

Maximum modulus estimates are obtained for generalized solutions of doubly nonlinear parabolic equations (DNPE). The equation \begin{equation*} \partial u/\partial t-\operatorname{div}\{|u|^l|\nabla u|^{m-2}\nabla u\}=0,\qquad m>1,\quad l>1-m, \tag{1} \end{equation*} is a prototype of a DNPE. Exact conditions on the parameters $m$ and $l$ are found that guarantee a local $L_\infty$-estimate for generalized solutions of Eq. (1), namely, \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*} Global maximum modulus estimates for generalized solutions of the first initial boundaty-value problem for a DNPE are given if the parameters $m$ and $l$ satisfy condition (2). Bibliography: 13 titles.