Hölder estimates for weak solutions of generate parabolic equations

A. V. Ivanov

Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 18, Tome 152 (1986), pp. 21-44

Résumé

We establish Hölder continuity, of solution of the degenerate parabolic equation $$ \frac{\partial u}{\partial t}-\operatorname{div}\vec a(x,t,u,\nabla u)+b(x,t,u,\nabla u)=0, $$ where $\vec a$ and $b$ are required to satisfy following conditions: \begin{gather*} |\vec a(x,t,u,\nabla u)|\leqslant\alpha_0|u|^{2\sigma}|p|+f_1(x,t),\\ \vec a(x,t,u,\nabla u)\cdot p\geqslant\nu_0|u|^{2\sigma}|p|^2+f_2^2(x,t),\\ |b(x,t,u,\nabla u)|\leqslant\beta_0|u|^{2\sigma}|p|^2+f_3^2(x,t), \end{gather*} $\sigma\geqslant0$, $\nu_0>0$, $\alpha_0$, $\beta_0\geqslant0$, $f_i\in L_{q,q_0}(Q_T)$, $i=1,2,3$ with appropriate exponents $q$, $q_0$. Interior estimates and estimates near the boundary of Holder exponents are obtained. Ho assumptions have been made concerning the sign of the solution.