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Hölder estimates for weak solutions of generate parabolic equations
A. V. Ivanov
Abstract:
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.
UDC:
517.946