An estimate for the modulus of continuity of generalized solutions of certain singular parabolic equations

A. V. Ivanov

Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 17, Tome 147 (1985), pp. 49-71

Résumé

One considers singular parabolic "equations’’ of the form $\partial\beta(u)/\partial t-\Delta u\ni0$, where $\beta(u)=a_0|u|^\lambda\operatorname{sign}u+\nu_0\operatorname{sign}u$, $a_0\geq0$, $\lambda>0$, $\nu_0\geq0$, $a_0+\nu_0>0$, $\operatorname{sign}u$ is a multivalued function, equal to $-I$ for $u0$, to $I$ for $u>0$, and to the segment $[-I,I]$ for $u=0$. Such a class of equations contains, in particular, the model for the two-phase Stefan problem, the porous medium equation, and the plasma equation. For the bounded generalized solutions $u(x,t)$ of the indicated equations (without the assumption $\partial u/\partial t\in L^2(Q_T)$ one has established a qualified local estimate of the modulus of continuity.