The Cauchy Problem for Quasilinear Parabolic Equations in the Degerate Case

Yu. N. Blagoveščenskiǐ

Teoriâ veroâtnostej i ee primeneniâ, Tome 9 (1964) no. 2, pp. 378-382

Résumé

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.