The Cauchy Problem for Quasilinear Parabolic Equations in the Degerate Case

In this paper we consider the differential properties of the solution to the Cauchy problem for the quasilinear parabolic equation begin{equation} \qquad\frac{\partial v}{\partial t}=\frac12\sum_{i,j=1}^n c_{ij}(t,x,v)\frac{\partial^2v}{\partial x_i\partial x_i}+\sum_{i=1}^na_i(t,x,v)\frac{\partial v}{\partial x_i}, \tag{1} \end{equation} where $c_{ij}=\sum_{k=1}^nb_{ik}(t,x,v)b_{jk}(t,x,v)$. Let class $C_T^{m,\gamma}$ be a class of continuous functions, which has bounded space derivatives up to the $m$-th order, and its $m$-th derivative is Hölder continuous with Hölder constant $\gamma$. It is proved in this paper that if
$$ \{b(t,x,v),a(t,x,v),\psi (x)\}\in C_T^{m,\gamma},\qquad m\geqq 2, $$
then $v\in C_{t_0}^{m,\gamma}$, where $t_0>0$ depends only on the constants of class $C_T^{m,\gamma}$. If $n=1$, $a(t,x,v)\equiv 0$, then the above assertion will follow for all $t\in [0,T]$ and $x\in(-\infty,\infty)$. It is noted that $b(t,x,v)$ may be any degenerate matrix. These assertions are proved by the method of diffusion processes.